Research Papers: Design Automation

Stability and Robustness Analysis of Uncertain Nonlinear Systems Using Entropy Properties of Left and Right Singular Vectors

[+] Author and Article Information
Young Kap Son

Department of Mechanical and
Automotive Engineering,
Andong National University,
1375 Gyeongdong-ro,
Andong-si, Gyeongsangbuk-do 36729, South Korea
e-mail: ykson@anu.ac.kr

Gordon J. Savage

Department of Systems Design Engineering,
University of Waterloo,
Waterloo, ON N2L 3G1, Canada
e-mail: gjsavage@uwaterloo.ca

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received May 5, 2016; final manuscript received November 25, 2016; published online January 12, 2017. Assoc. Editor: Xiaoping Du.

J. Mech. Des 139(3), 031402 (Jan 12, 2017) (11 pages) Paper No: MD-16-1340; doi: 10.1115/1.4035429 History: Received May 05, 2016; Revised November 25, 2016

This paper presents a novel approach to determine the stability space of nonlinear, uncertain dynamic systems that obviates the traditional eigenvalue approach and the accompanying linearizing approximations. In the new method, any long-term dynamic uncertainty is used in an extremely simple and economical way. First, the variability of the design variables about a particular design point is captured through the design of experiments (DOE). Then, corresponding computer simulations of the mechanistic model, over only a small time span, provide a matrix of discrete time responses. Finally, singular value decomposition (SVD) separates out parameter and time information and the expected uncertainty of the first few left and right singular vectors predicts any instability that might occur over the entire life-time of the dynamics. The singular vectors are viewed as random variables and their entropy leads to a simple metric that accurately predicts stability. The stable/unstable spaces are found by investigating the overall design space using an array of grid points of suitable spacing. The length of the time span needed to capture the nature of the dynamics can be as short as two or three periods. The robustness of the stability space is related to the tolerances assigned to the design variables. Errors due to sampling size, time increments, and number of significant singular vectors are controllable. The method can be implemented with readily available software. A study of two practical engineering systems with different distributions and tolerances, various initial conditions, and different time spans shows the efficacy of the proposed approach.

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Fig. 2

Stable system responses

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Fig. 3

An unstable system

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Fig. 4

First left singular and right singular vectors of stable and unstable systems: (a) U1 and (b) Q1

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Fig. 6

An unstable system

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Fig. 7

An apparent unstable system

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Fig. 8

Responses over a short time showing an indeterminate system

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Fig. 9

Schematic of vibration machine

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Fig. 5

Typical histogram for any singular vector

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Fig. 10

S-F space for vibration machine: (a) proposed method and (b) sector nonlinearity method [3]

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Fig. 11

Displacement responses for m1 at two different disputed design points: (a) [k, l] = [3, 3.9] and (b) [3, 3.2]

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Fig. 12

Schematic of the rotor system with only one bearing shown

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Fig. 13

S-F space with uniform distributions, 5% tolerances, and various time spans: (a) [0, 20 s], (b) [0, 15 s], (c) [0, 10 s], and (d) [0, 5 s]

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Fig. 14

Responses of main mass for uniform distributions and 5% tolerances: (a) [mb,m] = [1.5 kg, 5 kg], (b) [1.5 kg, 2 kg], (c) [2 kg, 5 kg], (d) [2.5 kg, 5 kg], (e) [3 kg, 2 kg], and (f) [5kg, 5 kg]

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Fig. 15

Stability spaces with time span [0, 15 s], uniform distributions and tolerances of (a) 10% and (b) 20%

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Fig. 16

S-F space using normal distributions with 20% statistical tolerance and time span [0, 15 s]

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Fig. 17

Transient responses of augmented system at [3 kg, 5kg]

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Fig. 18

S-F spaces for timespans: (a) 50 s, (b) 30 s, and (c) 15 s using uniform distributions and 5% tolerances



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