Flexible Spacecraft Dynamics, Control and Guidance by Leonardo Mazzini

Author:Leonardo Mazzini
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.64)

It follows that , and also that

(5.65)

So, as we wanted to demonstrate: , being the minimal distance of all the eigenvalues of from the critical point .

It is well known that where the are considered with their multiplicity.

If the system is stable, and we multiply the open loop transfer L by a factor k, we have .

So, as we wanted to demonstrate: it follows that and the closed loop system remains stable after the application of the multiplicative factor k.

The gain margin of the MIMO is the minimum real amplification factor which brings an eigenvalue to cross the critical point . The phase margin of the MIMO is the minimum rotation such that brings an eigenvalue to pass across the critical point . When an eigenvalue of L crosses the critical point the complete system becomes unstable.

The complementary sensitivity T is such that as can be easily verified from the definition given by Eq. 5.60. This means that when we have a low sensitivity the complementary matrix must be close to the unit matrix.

From Eq. 5.61 . We can use this property to guarantee the stability versus multiplicative perturbations of the primitive plant. Reusing the scheme of Fig. 5.2 on the G(s) of Fig. 5.4 any multiplicative perturbation of the plant G such that ensures that:

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