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Research Papers

Parametric Design Optimization of Uncertain Ordinary Differential Equation Systems

[+] Author and Article Information
Joe Hays

Control Systems Branch,Spacecraft Engineering Division,  Naval Center for Space Technology,  U.S. Naval Research Laboratory, Washington, DC 20375joehays@vt.edu

Adrian Sandu

Computational Science Laboratory, Computer Science Department,  Virginia Tech, Blacksburg, VA 24061sandu@cs.vt.edu

Corina Sandu

Advanced Vehicle Dynamics Laboratory, Mechanical Engineering,  Virginia Tech, Blacksburg, VA 24061csandu@vt.edu

Dennis Hong

Robotics and Mechanisms Laboratory, Mechanical Engineering,  Virginia Tech, Blacksburg, VA 24061dhong@vt.edu

J. Mech. Des 134(8), 081003 (Jul 23, 2012) (14 pages) doi:10.1115/1.4006950 History: Received June 06, 2011; Accepted May 15, 2012; Published July 23, 2012; Online July 23, 2012

This work presents a novel optimal design framework that treats uncertain dynamical systems described by ordinary differential equations. Uncertainty in multibody dynamical systems comes from various sources, such as system parameters, initial conditions, sensor and actuator noise, and external forcing. The inclusion of uncertainty in design is of paramount practical importance because all real-life systems are affected by it. Designs that ignore uncertainty often lead to poor robustness and suboptimal performance. In this work, uncertainties are modeled using generalized polynomial chaos and are solved quantitatively using a least-square collocation method. The uncertainty statistics are explicitly included in the optimization process. Systems that are nonlinear have active constraints, or opposing design objectives are shown to benefit from the new framework. Specifically, using a constraint-based multi-objective formulation, the direct treatment of uncertainties during the optimization process is shown to shift, or off-set, the resulting Pareto optimal trade-off curve. A nonlinear vehicle suspension design problem, subject to parametric uncertainty, illustrates the capability of the new framework to produce an optimal design that accounts for the entire family of systems within the associated probability space.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

An idealized 2-DOF deterministic quarter-car suspension model with a nonlinear asymmetric damper

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Figure 2

A representative road input signal created with a series of isolated speed bumps with filtered noise superimposed

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Figure 3

An uncertain 2-DOF quarter-car suspension model with a nonlinear asymmetric damper. The five uncertain parameters are, θ(ξ) = {ms (ξ), ks (ξ), bs (ξ), η(ξ), ku (ξ)}.

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Figure 4

A single 2D plane from the 3D Pareto optimal set showing the trade-off between the objective Ride and Rattle constraint; the Holding constraint is held constant, J¯Holding=0.034 [m]. Both the deterministic (dOpt) and uncertain (uOpt) cases are shown. These results confirm that the presence of uncertainty requires an off-set of the Pareto optimal solution set to realize a more robust design. The set enclosed by the ellipse correspond to Fig. 6.

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Figure 5

A single 2D plane from the 3D Pareto optimal set showing the trade-off between the objective Ride and Holding constraint; both the compression and extension Rattle constraints are held constant, J¯Rattle=J¯Rattle=0.203 [m]. Both the deterministic (dOpt) and uncertain (uOpt) cases are shown. These results confirm that the presence of uncertainty requires an off-set of the Pareto optimal solution set to realize a more robust design.

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Figure 6

Projection of the 3D deterministic and uncertain solutions onto the three orthogonal 2D planes. This is an example of an optimal solution with an active Rattle constraint. The constraint bounds are: J¯Rattle=J¯Rattle=0.203 [m] and J¯Holding=0.034 [m].

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Figure 7

The 2D projection of Fig. 5’s 3D deterministic and uncertain solutions onto the Holding/Rattle plane. This shows the transition from an active Holding constraint to an active Rattle constraint as the Holding bound is increased. The constraint bounds are: J¯Rattle=J¯Rattle=0.203 [m] and J¯Holding={0.0305,0.032,0.335,0.035} [m]. The markers for the deterministic and uncertain mean designs correspond within a given set. Also, the line of the uncertainty box associated with a given set matches the line of the Holding bound for that set.

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Figure 8

Parameter Pareto trade-off curve when J¯Rattle=J¯Rattle=0.203 [m]

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Figure 9

Monte Carlo results (1000 runs) showing 59.6% of the systems in the probability space violate the Rattle constraints when the deterministic optimal design is applied to an uncertain system; where J¯Rattle=J¯Rattle=0.152 [m] and J¯Holding=0.034 [m]

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Figure 10

Constraint violations from designs produced with the new framework can be controlled, or tuned, with the proper selection of the standard deviation scaling. Slightly increasing the scaling from a = 1 to a = 1.25 reduces the number of systems violating the constraints from 11.4% to 3.5%; where 1000 Monte Carlo simulations were used to determined the results; and J¯Rattle=J¯Rattle=0.152 [m] and J¯Holding=0.034 [m].

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