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The partial differential equations of motion for an uncontrolled distributed structure can be transformed into a set of independent modal equations by means of the system eigenfunctions.
In vibration analysis, the modal coordinates are referred to as natural coordinates. Active control forces generally recouple the modal equations so that the natural coordinates for the open-loop (uncontrolled) system cease to be natural coordinates for the closedloop (controlled) system. Control of this form is known as coupled control. In contrast, it is shown that a method known as the independent modal-space control method is a natural control method; i.e., the natural coordinates of the open-loop system and of the closed-loop system are identical. Furthermore, it is shown that natural control provides a unique and globally optimal closed-form solution to the linear optimal control problem for the distributed structure. The optimal control forces are ideally distributed. If implementation of distributed control is not feasible, then the distributed control forces can be approximated by finite-dimensional control forces. The class of self-adjoint systems are first considered following a treatment of non-self-adjoint systems. Numerical examples of a beam, a membrane and a whirling shaft are presented.
In general, the eigenquantities for a distributed structure cannot be computed in closed-form, so that spatial discretization of the differential eigenvalue problem is necessary. A common discretization method is the finite element method leading to a discrete eigenvalue problem. Two bracketing theorems characterizing convergence of the discrete eigenvalue problem derived by the finite element method to the differential eigenvalue problem are formulated.
The independent modal-space control method requires as many actuators as controlled modes. In contrast, coupled control is capable of controlling any number of modes using a single actuator, provided controllability is ensured. However, coupled control is sensitive to errors in the system parameters. As a compromise between coupled control and independent mbdal-space control, a block-independent control method is developed in which blocks of modes are controlled independently. The performances of independent modal-space control, coupled control and block-independent control are compared. |
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