3 INDEPENDENT MODAL SPACE CONTROL (IMSC) STRATEGY For a discrete set of measurements, x , the equation of motion for the 2-DOF lumped mass model of the laboratory structure shown in Fig. 1 can be determined as shown in Eq. 7. The parameters M C K , , are as shown in Eqs. 4 and 5. f denotes the vector of applied forces. For the IMSC approach, a decoupled independent modal description of a structure is often necessary. The decoupled independent modal space description outlined in Eq. 8, which is derived from Eq. 7 can be obtained from the transformation shown in Eq. 9 (Daley 2004, Inman 2001). φ is an orthonormal matrix of the eigenvectors of (1/2) (1/2) − −M KM . [ ]{ } [ ]{ } [ ]{ } f M x t C x t K x t = + + ( ) ( ) ( ) & && (7) mf +Λ +Ω = η η η & && (8) M x T 1/2 η φ = (9) Where: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Ω= ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ Λ= 2 2 1 1 1 , 2 2 i i i ω ω ς ω ς ω O O , fm- modal force, Λ - damping ratios, Ω - spectral matrix Thus, the problem is transformed from a MIMO (multiple-input multiple-output) control design problem in Eq. (7) into a series of multiple independent SISO (single-input single-output) control problem in Eq. (8). A general independent modal controller can now be defined as shown in Eq. (10). The closed loop description of each mode takes the form of Eq. (11), which enables the damping and frequency of each mode to be manipulated independently. The configuration of the vector of modal forces, mf can be set depending on the number of modes to be controlled whilst taking into account the number of sensors and actuators available as well. In the work presented here, two IMSC controllers are designed as explained in section 1.4. ( ) ( ) ( ) ( ) f s r s G s s m m m η = − (10) Where: ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ( ) 0 0 0 0 0 ( ) 0 0 ( ) 0 ( ) 2 1 g s g s g s G s m m M O M L L ( ) ( ) 2 1 ( ) , 2 2 r s g s s s s m i i i i i i + + + = ς ω ω η (11) Since the controller in Eq. (10) is defined in the modal space, it cannot be implemented directly and a transformation is necessary from the measured input signals. The matrix transformation required for extracting the first two bending modes of the laboratory structure at the locations pre-determined can be determined by making use of the rows of the matrix 1/2 MTφ . This enables Eq. (10) to be re-arranged in the physical domain as shown in Eq. (12) and this can now be implemented directly. A vector ( ) y s contains the sensor measurements and vector ( ) f s is the local demand force. ( ) ( ) ( ) ( ) 1/2 1/2 1/2 fsM rsM Gs Mys T m m φ φ φ − = (12) The local demand forces can now be determined from the linear displacements, velocities and accelerations at each actuator location as noted in Eq. (12). For the work presented in this paper, once the local demand forces are evaluated, an inverse actuator model (i.e. an inverse model for the APS Dynamics Model 400 Electrodynamic Shakers) is used to calculate the desired control voltage signal which is then transmitted to the actuators. A typical global processing stage is demonstrated in Fig. 3. 218
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