Controllability and Observability of the DiscreteTime Structural Model
Consider now a structure in modal coordinates. Similar to the continuous-time grammians the discrete-time grammians in modal coordinates are diagonally dominant, where Wci and W0i are 2 x 2 blocks, such that Wci wCiI2 and W0i w0iI2, see 98 , where IK II2 2 1 - cos fflj At 2 1 - cos fflj At wci TT--Y- wci cont-Yl-- 4.8 Q In the above equations Bmi is the ith block of Bm in modal coordinates, and Cmi is the ith block of Cm in modal coordinates, where Cm CmqQ_1 Cmv , see 2.42 for Zs0. In the...
Approximately Balanced Structure in Modal Coordinates
By comparing the grammians in the balanced and modal representations we noticed that the balanced and modal representations are close to each other. The closeness of the balanced and modal representations can also be observed in the closeness of the system matrix A in both representations. It was shown in Chapter 2 that the matrix A in modal coordinates is diagonal with a 2 x 2 block on the diagonal . We show that the system matrix A in the balanced representation is diagonally dominant. Figure...
TimeLimited Grammians
The steady-state grammians, defined over unlimited time integrals, are determined from the Lyapunov equations 4.5 . The grammians over a finite-time interval T t1, t2 where 0 lt t1 lt t2 lt lt are defined by 4.3 , and can be obtained from the matrix differential equations 4.4 . In many cases these equations cannot be conveniently solved, and the properties of their solutions are not readily visible. However, using the definitions from 4.3 we will derive the closed-form grammians over the finite...
Diagonally Dominant Grammians
Assuming small damping, such that 1, where max , i 1, , n, the balanced and modal representations of flexible structures are closely related. One indication of this relationship is expressed in the grammian form. The balanced grammians are equal and diagonal similarly, the grammians in modal coordinates are diagonally dominant, and by using appropriate scaling, they are approximately equal. This is expressed in the following property Property 4.1. Diagonally Dominant Grammians in Modal...
Sensor Placement Strategy
1. Actuator locations are already determined. 2. Select the areas where the sensors can be placed, obtaining the R candidate sensor locations. 3. Determine the sensor placement indices lt rk i for all the candidate sensor locations i 1, , R , and for all the modes of interest k 1, , n . 4. For each mode, select r1 for the most important sensor locations. The resulting number of sensors r2 for all the modes considered i.e., r2 lt n x r1 is much smaller than the number of candidate locations,...