By Wodek Gawronski Dr (auth.), Wodek Gawronski Dr (eds.)
Methods of structural keep watch over and dynamics are brought during this publication. those contain aid of enormous structural types by way of balanced truncation, placement of actuators and sensors for dynamic trying out and keep watch over, structural identity of the minimum-order balanced illustration, balanced dissipative controller layout, balanced LQG and H( controller designs with the closed-form relationships among controller parameters and process functionality, and controller relief equipment that guard the closed-loop functionality. The publication explores the original houses of versatile constructions to procure effective tools of dynamic research and controller layout. The provided equipment of structural dynamics, identity, sensor/actuator placement, and passive, LQG and H( controller layout were checked either with simulations and commercial implementations.
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Additional info for Balanced Control of Flexible Structures
Is the expectation operator. . 6) that the minimum of zx cannot be determined, unless the full combination of truncated states is exhausted (the difficulty lies in the determination of the covariance W~). 7) In the above equation Uyll2=tr(CrCr'2), and tly~u2=tr(Cr~c~2). Define a scalar product of vectors x and y as
61) holds. The above properties, which connect A, B, and C, and the grammians, are used later to derive additional properties of balanced flexible structures and controllers. 2 Closed-Form Grammians in Modal Coordinates In order to further investigate the connections between the flexible structures in the balanced and modal coordinates, a closed-form grammians are derived. , such that ~ i < l , and separated natural frequencies, ~oi, ~oi~-t~ j, for i*j, and i d = l , .... n. Assume also that if the natural frequencies occur in clusters of repeated frequencies, the modes related to these frequencies are mutually independent.
However, if parts of the states are weakly controllable and observable at the same time (weakly influenced by the plant input and weakly recorded in the plant output), they can be ignored in the controller design. The joint controllability and observability properties of the system are characterized by the Hankel singular values, and this simple idea was developed by Moore . Instead of using the controllability and observability properties, Skelton , [i02], , ,  characterizes each state variable with its cost.