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Research Papers

Dual Faceted Linearization of Nonlinear Dynamical Systems Based on Physical Modeling Theory

[+] Author and Article Information
H. Harry Asada

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
MIT Room 3-346,
Cambridge, MA 02139
e-mail: asada@mit.edu

Filippos E. Sotiropoulos

Department of Mechanical Engineering,
Massachusetts Institute of Technology,
77 Massachusetts Avenue,
MIT Room 3-346,
Cambridge, MA 02139

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received December 24, 2017; final manuscript received August 28, 2018; published online October 5, 2018. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 141(2), 021002 (Oct 05, 2018) (11 pages) Paper No: DS-17-1631; doi: 10.1115/1.4041448 History: Received December 24, 2017; Revised August 28, 2018

A new approach to modeling and linearization of nonlinear lumped-parameter systems based on physical modeling theory and a data-driven statistical method is presented. A nonlinear dynamical system is represented with two sets of differential equations in an augmented space consisting of independent state variables and auxiliary variables that are nonlinearly related to the state variables. It is shown that the state equation of a nonlinear dynamical system having a bond graph model of integral causality is linear, if the space is augmented by using the output variables of all the nonlinear elements as auxiliary variables. The dynamic transition of the auxiliary variables is investigated as the second set of differential equations, which is linearized by using statistical linearization. It is shown that the linear differential equations of the auxiliary variables inform behaviors of the original nonlinear system that the first set of state equations alone cannot represent. The linearization based on the two sets of linear state equations, termed dual faceted linearization (DFL), can capture diverse facets of the nonlinear dynamics and, thereby, provide a richer representation of the nonlinear system. The two state equations are also integrated into a single latent model consisting of all significant modes with no collinearity. Finally, numerical examples verify and demonstrate the effectiveness of the new methodology.

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Figures

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Fig. 1

Conceptual diagram of DFL. A nonlinear dynamical system is viewed from an augmented space consisting of independent state variables and auxiliary variables, which are nonlinearly related to each other.

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Fig. 2

Example of bond graph

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Fig. 3

Examples of constitutive laws of elements: (a) resistor R1, (b) capacitor C, and (c) inertia I1

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Fig. 4

Algebraic loop involved in the bond graph in Fig. 2

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Fig. 5

Example of causal auxiliary variables where input u(t) is not involved in each causal path

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Fig. 6

Example of an anti-causal auxiliary variable where input u(t) is involved in the causal path. Backtracking the causal path terminated at auxiliary variable f leads to input u(t).

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Fig. 7

Bond graph of a dynamical system inspired by an excavator powered by a hydraulic system

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Fig. 8

Constitutive laws of the system in Fig. 7

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Fig. 9

Comparison of three linearization methods, Taylor expansion, statistical linearization, and DFL, in terms of the root mean square of predicting η˙ . The bar chart is normalized by the root-mean-square (RMS) of DFL: σDFL .

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Fig. 10

Schematic of deviated trajectories of linearized models from the exact nonlinear model

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Fig. 11

Deviation envelopes of different linearization models. The maximum deviation from the exact nonlinear model over the time horizon of 0.3 s is shown by the envelope.

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Fig. 12

Sum of mean squared errors of all the state variables. Comparison of the three linearization models against time horizon.

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Fig. 13

Bond graph of a system with anti-causal auxiliary variable

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Fig. 14

Deviation envelopes of simulations using the causal auxiliary variables η*

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