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Research Papers

# A Higher-Order Method for Dynamic Optimization of Controllable Linear Time-Invariant Systems

[+] Author and Article Information
Damiano Zanotto

Mechanical Systems Laboratory,
Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716

Giulio Rosati

Department of Management
and Engineering (DTG),
University of Padua—Faculty of Engineering,
via Venezia 1,
35131 Padova, Italy
e-mail: giulio.rosati@unipd.it

Sunil K. Agrawal

Mechanical Systems Laboratory,
Department of Mechanical Engineering,
University of Delaware,
Newark, DE 19716
e-mail: agrawal@udel.edu

For each i, start from the last column $i+(γi)m$ and proceed backward to the left, noticing that each new column has a 1 where all the previous columns have a zero. Also, the pattern of the null entries of each selected column prevents it from being a linear combination of columns corresponding to a different i.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received March 21, 2011; final manuscript received August 1, 2012; published online November 7, 2012. Assoc. Editor: Rama K. Yedavalli.

J. Dyn. Sys., Meas., Control 135(2), 021008 (Nov 07, 2012) (10 pages) Paper No: DS-11-1125; doi: 10.1115/1.4007707 History: Received March 21, 2011; Revised August 01, 2012

## Abstract

This work describes a new procedure for dynamic optimization of controllable linear time-invariant (LTI) systems. Unlike the traditional approach, which results in 2 n first-order differential equations, the method proposed here yields a set of m differential equations, whose highest order is twice the controllability index of the system p. This paper generalizes the approach presented in a previous work to any controllable LTI system.

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## References

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## Figures

Fig. 2

Values of the functional J for increasing k. The top plot refers to system 1, whereas the bottom plots refers to system 2.

Fig. 3

Normalized state trajectories for system 1. Dashed lines represent the trajectories of the zero mode solution (i.e., k = 0), solid lines represent the optimal solutions for increasing values of k=4,5,…,10.

Fig. 4

Normalized state trajectories for system 2. Dashed lines represent the trajectories of the zero mode solution (i.e., k=0), solid lines represent the optimal solutions for increasing values of k=7,…,10.

Fig. 1

6-DOF spring-mass-damper system. In the examples presented here, only three masses are directly actuated.

## Errata

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