Active damping with collocated pairs

Abstract

The role of damping in the gain stabilization of a control system in the roll-off region has been pointed out in the foregoing chapter. The damping also reduces the settling time of the transient response to impulsive loads. Indeed, since the modal expansion of the impulse response matrix corresponding to (2.19) is

$$ g\left( \mathcal{T} \right) = \sum\limits_{i = 1}^n {\frac{{{\emptyset _i}\emptyset _i^T}}{{{\mu _i}{\omega _{di}}}}} {e^{ - {\xi _i}{\omega _i}\mathcal{T}}}\sin {\omega _{di}}\mathcal{T} $$

(5.1)

[g(τ) and G(ω) are a Fourier transform pair, see problem P.2.6], one readily sees that the time constant (the memory) of each modal contribution is proportional to τ _i ∼ (ξ _i ω _i )^-1. If, for example, ω _i ≃ 1 rad/s and ξ _i = 0.002, which are common values for spacecrafts, the time to reduce the impulse response by a factor of 10 is longer than 1000 s, comparable to that of one orbit revolution of the spacecraft. If one wants, for example, to maintain a micro-gravity environment or the pointing of a telescope, in spite of the transient loads from the thrusters or the human activity, one easily appreciates the need for damping augmentation. Similarly, the damping reduces the amplitude of the frequency response functions in the vicinity of the resonances and, as a result, the steady state response to wide-band disturbances [the variance of the stationary modal response to a white noise excitation is proportional to ξ ^-1 _i ω ^-3 _i ].