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Research Papers

# Passive Mechanical Control With a Special Class of Positive Real Controllers: Application to Passive Vehicle Suspensions

[+] Author and Article Information
Michael Z. Q. Chen

Department of Mechanical Engineering,
The University of Hong Kong,
Hong Kong
e-mail: mzqchen@hku.hk

Yinlong Hu

School of Automation,
Nanjing University of Science and Technology,
Nanjing 210094, China

Fu-Cheng Wang

Department of Mechanical Engineering,
National Taiwan University,
Taipei 10617, Taiwan

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 24, 2015; final manuscript received September 10, 2015; published online October 6, 2015. Assoc. Editor: Douglas Bristow.

J. Dyn. Sys., Meas., Control 137(12), 121013 (Oct 06, 2015) (11 pages) Paper No: DS-15-1193; doi: 10.1115/1.4031630 History: Received April 24, 2015; Revised September 10, 2015

## Abstract

This paper presents an efficient H2 optimization method for passive mechanical control problems with a special class of positive real controllers. In particular, the problem of designing passive vehicle suspensions based on a full-car model is taken as an example, where both the positive real constraint and the constraint imposed on the static stiffness are considered. An unconstrained nonlinear programming problem is formulated by using the structured H2 optimization framework, and the Lagrange matrix multiplier method is employed to derive a set of necessary conditions for the optimization so that the time-efficient gradient-based algorithms can easily be implemented. The proposed method can also effectively deal with the fixed static stiffness optimization problem and it is shown in the numerical examples that the proposed method cannot only recover the existing fixed-structure configuration, but also introduce new (optimal) configurations with respect to the specific weighting factors, which demonstrates the effectiveness of the proposed method.

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## Figures

Fig. 2

Control synthesis paradigm with a special class of positive real controller

Fig. 3

A seven degrees-of-freedom full-car model

Fig. 1

Circuit symbols and correspondences with definition equations and admittance Y(s) [1]

Fig. 4

Static output feedback formulation with a structured feedback gain

Fig. 5

The conventional passive strut, denoted as C1

Fig. 6

The series configuration, denoted as C2

Fig. 7

The newly obtained configuration, denoted as C3

Fig. 8

Performance comparison with respect to different road profiles and different velocities: optimal C1 (circle), the optimal special class admittance (solid), and C3 for V = 30 m/s and class C road (dash)

Fig. 9

Optimal ride comfort performance J1 (left) and the percentage improvement over the conventional configuration C1 (right): the special class admittance in this paper (solid) and the conventional configuration C1 (dash)

Fig. 10

Optimal road holding performance J3 (left) and the percentage improvement over the conventional configuration C1 (right): the special class admittance in this paper (solid) and the conventional configuration C1 (dash)

Fig. 11

The suspension configurations for optimal ride comfort performance when kf/kr=80 kN/m

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