On the Calculation of Eigenvalues and Eigenvectors of Matrix Polynomials and Orthogonality Relations between their Eigenvectors

In the paper, the computation of the eigenvalues and eigenvectors of polynomial eigenvalue problem via standard eigenvalue problems is presented. We also establish orthogonality relations between the eigenvectors of matrix polynomials. A numerical example is given to illustrate the applicability of the obtained theoretical results

Optimal actuation in active vibration control using pole-placement

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

Restoring dynamic properties of damaged buildings using pole assignment control method

In this paper pole assignment control method is used to modify dynamic properties of damaged buildings. Since the eigenvalues of open-loop system, known as poles, are related to the dynamic properties of the system, damage causes changes in their values. In terms of pole assignment, a controller will modify dynamic properties of damaged structure by moving the eigenvalues to their initial place. Using the proposed gain matrix calculation algorithm and restoring the initial eigenvalues, dynamic properties of undamaged building, which have been lost due to damage, can be restored. When feedback controller restores all the eigenvalues, damaged elements and connections can be verified from the large control forces concentrated in the nodes of damaged elements. The algorithm also represents a new approach for partial pole assignment and the effect of restoring some undesired eigenvalues can be studied individually. For numerical simulation, a typical ten-story moment frame building is employed which has experienced some damages during the 1971 Newhall earthquake. Three states of damage are defined for the finite element model of the building to consider damage in elements and connections

Closed form solutions to the optimality equation of minimal norm actuation

This research focused on the problem of minimal norm actuation in the context of partial natural frequency or pole assignment applied to undamped vibrating systems by state feedback control. The result of the research was the closed form solutions for the minimal norm control input and gain vectors. These closed form solutions should took open loop eigenpairs and the desired frequencies of the controlled system and outputted the optimal controller parameters. This optimization technique ensures that the system’s dynamics will be effectively controlled while keeping the controller effort minimal. The controller must then be able to shift only the desired the system poles anywhere in the complex s-plane in order to give the system certain desired characteristics with no spillover. The open loop system dynamics were found by applying a discrete model of the studied vibrating system and then finding the eigenvalue problem associated with the second-order open loop system equations. A first order realization was then performed on the system in order to know its response to certain initial conditions. The system’s dynamics where to be modified via closed loop control. Partial natural frequency assignment was chosen as the control technique so that certain system frequencies could be left untouched to ensure that the system will not respond in an unexpected manner. The control was to be optimized by minimizing the norm of the control input and gain vectors. A closed form solution for these vectors was found in so that these vectors could be simply calculated using an algorithm that takes the open loop eigenpairs and the desired eigenvalues of the system and outputs the two vectors. This closed form solution was successful implemented and the controller parameters found were applied to a vibrational system. A simulation for the un-optimized and optimized cases was performed applying both controllers to the same system. The response and controller forces for both cases were plotted in MATLAB and compared. Both systems showed the desired system response meaning that they both had the same effect on the system. Inspecting both controller efforts showed that the optimal control case simulation showed less controller effort than the arbitrary case thus showing successful implementation of minimal norm actuation

Application of a finite element model update technique to detect damage in a beam structure

The interest in the ability to monitor a structure and detect damage at the earliest possible stage is pervasive throughout the civil, mechanical and aerospace engineering communities. The thesis focuses on the application of a finite element model updating technique to monitor and detect damage in beams. A Sensitivity Based Element-by- Element (SBEBE) methodology is chosen as the finite element model updating technique. In this method, damage is detected by updating a finite element model with test data obtained from healthy and damaged structures and observing the relative changes in the updated finite element models. The performance, efficiency and sensitivity of the SBEBE algorithm in detecting the damage location and severity are studied through numerical and experimental test cases on a cantilever beam. The location and extent of damage are successfully predicted with all numerical cases. The presence of noise in the numerical data and its effects on the damage detection process are examined. The SBEBE algorithm is capable of detecting the presence, location and extent of damage for noise levels in the numerically generated data up to 5% of the signal amplitude. Also experimental studies are carried out on a cantilever beam with modal data measured using a laser doppler vibrometer. A small section of the cantilever beam is mechanically removed, and the SBEBE algorithm is used successfully to detect the damage location and severity --Abstract, page iii