Type of Document Dissertation Author Guzzardo, Carla Ann Author's Email Address cguzzard@tigers.lsu.edu, carla.guzzardo@gmail.com URN etd-10302012-133755 Title Optimal Actuation in Active Vibration Control Using Pole-Placement Degree Doctor of Philosophy (Ph.D.) Department Mechanical Engineering Advisory Committee

Advisor Name Title Pang, Su-Seng Committee Co-Chair Ram, Yitshak M. Committee Co-Chair de Queiroz, Marcio Committee Member Li, Guoqiang Committee Member de Hoop, Cornelis Dean's Representative Keywords

- vibration
- state feedback control
- pole assignment
- partial pole placement
- resonance avoidance
- minimization of control effort
- optimal actuation
- active vibration control
Date of Defense 2012-10-10 Availability unrestricted AbstractThe purpose of this study was to find and demonstrate a method of optimal actuation in amechanical 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.

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