Research Papers

Pole Placement for Single-Input Linear System by Proportional-Derivative State Feedback

[+] Author and Article Information
Taha H. S. Abdelaziz

Department of Mechanical Engineering,
Faculty of Engineering,
Helwan University,
Helwan, Cairo 11792, Egypt
e-mail: tahahelmy@helwan.edu.eg

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 3, 2014; final manuscript received September 28, 2014; published online November 7, 2014. Assoc. Editor: M. Porfiri.

J. Dyn. Sys., Meas., Control 137(4), 041015 (Apr 01, 2015) (10 pages) Paper No: DS-14-1160; doi: 10.1115/1.4028713 History: Received April 03, 2014; Revised September 28, 2014

This paper deals with the direct solution of the pole placement problem for single-input linear systems using proportional-derivative (PD) state feedback. This problem is always solvable for any controllable system. The explicit parametric expressions for the feedback gain controllers are derived which describe the available degrees of freedom offered by PD state feedback. These freedoms are utilized to obtain closed-loop systems with small gains. Its derivation is based on the transformation of linear system into control canonical form by a special coordinate transformation. The solving procedure results into a formula similar to Ackermann’s one. In the present work, both time-invariant and time-varying linear systems are treated. The effectiveness of the proposed method is demonstrated by the simulation examples of both time-invariant and time-varying systems.

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Grahic Jump Location
Fig. 3

Zero-input transient response of open-loop system

Grahic Jump Location
Fig. 2

Transient response and control input of a LTI system controlled by PD feedback

Grahic Jump Location
Fig. 4

Gain elements of a LTV system controlled by PD state feedback when F = (1, 2, 3)

Grahic Jump Location
Fig. 5

Gain elements of a LTV system controlled by PD state feedback when F = (0.0026, −0.1352, 1.00001)

Grahic Jump Location
Fig. 6

Transient response and control input of a LTV system controlled by PD state feedback

Grahic Jump Location
Fig. 7

Gain elements of a LTV system controlled by state feedback

Grahic Jump Location
Fig. 8

Gain elements of a LTV system controlled by state-derivative feedback



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