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Chen System As a Controlled Weather Model - Physical Principle, Engineering Design and Real Applications

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Chen System As a Controlled Weather Model - Physical Principle, Engineering Design and Real Applications Chen System as a Controlled Weather Model - Physical Principle, Engineering Design and Real Applications Sooraksa, Pitikhate; Chen, Guanrong Published in: International Journal of Bifurcation and Chaos Published: 01/04/2018 Document Version: Final Published version, also known as Publisher’s PDF, Publisher’s Final version or Version of Record License: CC BY Publication record in CityU Scholars: Go to record Published version (DOI): 10.1142/S0218127418300094 Publication details: Sooraksa, P., & Chen, G. (2018). Chen System as a Controlled Weather Model - Physical Principle, Engineering Design and Real Applications. International Journal of Bifurcation and Chaos, 28(4), [1830009]. https://doi.org/10.1142/S0218127418300094 Citing this paper Please note that where the full-text provided on CityU Scholars is the Post-print version (also known as Accepted Author Manuscript, Peer-reviewed or Author Final version), it may differ from the Final Published version. When citing, ensure that you check and use the publisher's definitive version for pagination and other details. General rights Copyright for the publications made accessible via the CityU Scholars portal is retained by the author(s) and/or other copyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associated with these rights. Users may not further distribute the material or use it for any profit-making activity or commercial gain. Publisher permission Permission for previously published items are in accordance with publisher's copyright policies sourced from the SHERPA RoMEO database. Links to full text versions (either Published or Post-print) are only available if corresponding publishers allow open access. Take down policy Contact [email protected] if you believe that this document breaches copyright and provide us with details. We will remove access to the work immediately and investigate your claim. Download date: 23/09/2021 April 24, 2018 8:15 WSPC/S0218-1274 1830009 International Journal of Bifurcation and Chaos, Vol. 28, No. 4 (2018) 1830009 (12 pages) c The Author(s) DOI: 10.1142/S0218127418300094 Chen System as a Controlled Weather Model — Physical Principle, Engineering Design and Real Applications Pitikhate Sooraksa Department of Computer Engineering, Faculty of Engineering, King Mongkut’s Institute of Technology, Ladkrabang, Bangkok 10520, Thailand [email protected] Guanrong Chen∗ Department of Electronic Engineering, City University of Hong Kong, P. R. China [email protected] Received January 18, 2018; Revised March 19, 2018 This paper presents the Chen system as a controlled weather model. Mathematically, the Chen system is dual to the Lorenz system via time reversal. Physically, the Chen system can be viewed as a controlled weather model from the anti-control perspective. This paper illustrates the physical principle of this controlled weather model, and develops an engineering design of the model for real indoor climate (temperature-humidity) regulation, with a perspective on outdoor weather control application. Keywords: Chaos; Chen system; Lorenz system; temperature-humidity control; climate control. 1. Introduction where typical parameters are c = 28, a = 10, and b =8/3. The so-called “anti-control of chaos” was The Chen system [Chen & Ueta, 1999] is described Int. J. Bifurcation Chaos 2018.28. Downloaded from www.worldscientific.com meant to introduce a control input to make an by originally nonchaotic system become chaotic [Chen, x a y − x , ˙ = ( ) 1997]. Here, a simple linear anti-controller of the u αx βy γz y˙ =(c − a)x − xz + cy, (1) form = + + was applied to the right- hand side of the second equation of the Lorenz equa- z˙ = xy − bz, tion (2), which had the three parameters originally where, to generate chaos, typical parameters are not in the chaotic region, to try to generate (new) c = 28, a = 35, and b = 3. It was originally chaotic dynamical behaviors. Based on the well- by CITY UNIVERSITY OF HONG KONG on 01/22/19. Re-use and distribution is strictly not permitted, except for Open Access articles. derived from anti-control of the classical Lorenz sys- known criterion of Shilnikov [1965], it was found tem [Lorenz, 1963] that the anti-controller could be chosen as x˙ = a(y − x), y˙ = cx − xz − y, u = −ax +(1+c)y +0z (3) (2) z˙ = xy − bz, thereby yielding the Chen system (1). ∗ Author for correspondence This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. 1830009-1 April 24, 2018 8:15 WSPC/S0218-1274 1830009 P. Sooraksa & G. Chen (a) (b) Fig. 1. Comparison of two chaotic attractors. (a) Chaotic Chen attractor and (b) chaotic Lorenz attractor. As a side note, by similarly choosing u = the Chen system (1) can be interpreted as follows −cx +(1+c)y +0z, one obtains the Lu system [Rohsenow et al., 1985]: [Lu & Chen, 2002] • x is a variable of the spatial average of hydro- x˙ = a(y − x), y˙ = −xz + cy, dynamic velocity, which is proportional to the (4) z xy − bz, intensity of the fluid convective motion; ˙ = • y is a variable of temperature, which is the tem- where typical parameters are c =22.2, a = 30, and perature difference between the ascending and b =44/15. descending currents; • z By comparing the chaotic attractors of the is a variable of the temperature gradient or Chen system to the Lorenz system, generated with deviation, which is proportional to the distortion of the vertical temperature profile from linearity; the same initial condition x0 = −3, y0 =2, • dimensionless positive constant a is the Prandtl z0 = 20 as shown by Fig. 1, the new attrac- tor moves upward much more fiercely, prominently number; • c driven by the internal force governed by the new dimensionless positive constant is the Rayleigh input (3) to the system equations. This paper illus- number, which is the product of the Grashof a trates the physical principle and mechanism that number and the Prandtl number, therefore and c create this difference in their chaotic dynamics, and are mutually dependent; • b explains why some seemingly simple modifications dimensionless positive constant is related to the of the Lorenz system provide drastically different ratio of the spatial dimensions involved. dynamics. It will be further shown that the new Recall that the Prandtl number (Pr) is the ratio dynamical mechanism embedded in the Chen sys- of momentum diffusivity to thermal diffusivity. A tem, which by nature is a controlled Lorenz sys- smaller Pr value implies a more rapid heat diffu- tem, could be useful as an engineering design for sion compared to the momentum, whereas a larger Int. J. Bifurcation Chaos 2018.28. Downloaded from www.worldscientific.com some real applications, with two examples presented Pr value means that the momentum becomes more on indoor climate (temperature-humidity) regula- diffusivity dominant on the dynamics. Recall also tion and a perspective on outdoor weather control that the Grashof number is defined as the ratio of application. the buoyancy force to the viscous force acting on the The rest of the paper is organized as follows. fluid, which often arises in the study of free convec- Section 2 illustrates the physical principle of the tion or natural convection. And, the Rayleigh num- Chen system developed from the Lorenz system. ber can be described by the product of these two by CITY UNIVERSITY OF HONG KONG on 01/22/19. Re-use and distribution is strictly not permitted, except for Open Access articles. Section 3 develops an engineering design utilizing ratios. In the atmospheric convection model, the the new characteristics of the Chen system. Sec- Rayleigh number is proportional to the difference tion 4 shows two examples of indoor and outdoor in the temperatures from the warm base to the cool climate controls. Section 5 concludes the paper with top in a convective cell. Unlike the Prandtl number, an outlook on potential future applications. both the Grashof number and the Rayleigh number depend on the differences between the surface tem- perature and the bulk temperature, as well as on a 2. Physical Interpretation characteristic length in the model. Similarly to the Lorenz system, which is a simpli- Although the variables and parameters in fied weather dynamics model, the parameters in Eqs. (1)–(3) are dimensionless after reasonable 1830009-2 April 24, 2018 8:15 WSPC/S0218-1274 1830009 Chen System as a Controlled Weather Model mathematical simplifications, their corresponding the controller (3), which is a fundamental property physical properties are preserved, which are fur- of the Chen system distinguishing itself from the ther discussed below. Specifically, in the following, Lorenz system. variable x is considered as the air flow, y as the More precisely, keeping in mind the magnitudes temperature, and z as the relative humidity. The of the parameters and signs of the variables in the z variable represents a cross-coupling effect and is control input (3), the first term −ax can be inter- inherently a fluctuated variable of the temperature. preted as reversing the direction of the air flow, Usually, temperature affects humidity, namely the which reduces the rate of temperature change along water vapor or humidity is limited by the temper- the x-axis scaled by a factor of a. The second term ature in a parcel of air. In fact, humidity responds (1 + c)y can be interpreted as modifying the tem- to temperature in such a way that with higher perature by adding more heat flux to the system, temperature the air holds more water vapor. Note, scaled by a factor of (1+c). The schematic diagram however, that humidity has a rather different tem- shown in Fig.
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