LQR based MIMO-PID controller for the vector control of an underdamped harmonic oscillator

Closed loop control of vibration field transient: Application to wave focusing

“…The zeros might be affected in a more complicated way, but if the modulation frequency is large enough compared to the dynamic of the oscillator, they roughly behave in a similar manner as the poles. Actually, in [16], the proposed method provides the equations of the demodulated before filtering. Assuming that the poles of η (k) are slow compared to Ω k (the modulation angular frequency) and the bandwidth of the controller is small, one can write in complex polar form:…”

“…( 7) is written directly in an analytical form as a function of the modal parameters ξ k , ω k and the modulation/demodulation frequency Ω k . By this way, a systematic approach for the controller design of a single mode has been proposed in [16], which is described in the next section.…”

“…Following [16], particular weighting matrices Q k and R k are chosen: R k = diag(q ki , q ki , q kp , q kp , 0, 0) (11)…”

“…The identification procedure consists in a frequency sweep around the resonance frequency of given controlled mode by using the associated sensors and actuators. The modal parameters are extracted using the quadrantal frequencies method as explained in [16]. It can be observed that all the controlled modes are lightly damped ξ k < 0.01 (1%…”

“…The paper is organized as follows. The section 2 recalls the modal dynamic model of one vibration mode in its demodulated base the design of a controller described in [16] and addresses the control of several modes. In section 3, the experimental device is presented and different design protocols are developed specifically.…”