# Optimal actuation in active vibration control using pole-placement

## Identifier

etd-10302012-133755

## Degree

Doctor of Philosophy (PhD)

## Department

Mechanical Engineering

Dissertation

## Abstract

The purpose of this study was to find and demonstrate a method of optimal actuation in a mechanical system to control its vibration response. The overall aim is to develop an active vibration control method with a minimum control effort, allowing the smallest actuators and lowest control input. Mechanical systems were approximated by discrete masses connected with springs and dampers. Both numerical and analytical methods were used to determine the optimum force selection vector, or input vector, to accomplish the pole placement, finding the optimal location of actuators and their relative gain so that the control effort is minimized. The problem was of finding the optimal input vector of unit norm that minimizes the norm of the control gain vector. The methods of pole placement and partial pole placement were introduced, and used to solve various problems, including the active natural frequency modification problem associated with resonance avoidance in undamped systems, and the single-input-multiple-output pole assignment problem for second order systems. Both full and limited controllability were addressed. During the numerical analysis, it was discovered that the system is uncontrollable if a control input vector is chosen that is mathematically orthogonal to an eigenvector associated with a reassigned eigenvalue. Conversely, the optimal input vector was discovered to be mathematically parallel to an eigenvector. This was proven analytically through mathematical proofs and demonstrated with various examples. Simulations were performed in MATLAB and Maple to verify the results numerically. An example using realistic units was developed to show the order of magnitude improvement expected by using this method of optimization. All initial conditions and system parameters were held the same, but the input vector was changed. The optimal input vector provided an order of magnitude improvement over an evenly distributed input vector. The principal conclusion was that by choosing a state feedback input vector that is mathematically parallel to the eigenvector associated with the open-loop eigenvalue to be reassigned, or in the case of multiple assignments, in the subspace of the eigenvectors, the control effort to accomplish pole placement can be reduced to its minimal value.

2012

## Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Pang, Su-Seng