APPLICATION OF THE IMPROVED ALGORITHM
The improved algorithm of subsection 5.2 is applied. Most of the frequency intervals, which correspond to the 50 points of the initial frequency gridding, were eliminated, and the basic algorithm of subsection 5.1 is applied only to 8 intervals. The same result is obtained for Umax – The computations were done in only 474 s (about 8 minutes). This improved solution appears thus especially computationally efficient. Moreover, the result is nearly non conservative.
6.1 A PHYSICAL INTERPRETATION OF THE ti LOWER BOUND
The aim of this subsection is to illustrate the usefulness of a q lower bound. Let (А, В, C, 0) a state-space representation of the transfer matrix M(s) in the interconnection structure M(s) – Д. A destabilizing perturbation Д* is provided with the fi lower bound. Remember that the value of this lower bound is 1.94 at 9.97 rad/s with the method by (Magni and Doll, 1997). This means that the norm of Д* is 1/1.94 and that the closed loop state matrix A + BA*Chas a pole on the imaginary axis at ± 9.97 rad/s.
This is confirmed by the root locus of Figure 10.5. This root locus
was obtained by plotting the eigenvalues of A + BaA*Cwhen a varies between 0 and 1. The star "*" represents the initial poles (associated to a = 0), while the circle "o" represents the final poles (associated to a = 1). The closed loop rigid poles at the bottom of the Figure are not especially moved by the application of the destabilizing perturbation Д*.
Root locus associated to the destabilizing model perturbation, which is
provided with the fi lower bound – the dotted lines each correspond to an isovalue of the damping ratio, namely 1 %, 2 %, … ,10 % and 10 %, 20 %, … ,100 %.
The 2 closed loop flexible modes, which are not actively controlled (see subsection 1.4 of chapter 2), are neither especially moved by the application of A*. Note that two poles are associated to each flexible mode in the figure: one for the state-feedback controller and one for the observer. These two poles do not coincide, even if they are close, because the state-feedback and observer gains were computed with two different methods.
The 4 closed loop flexible modes, which are actively controlled, are
moved indeed by the application of Д*. The closed loop flexible pole, which crosses the imaginary axis, corresponds to the state-feedback closed loop pole:
-2.59Є+00 + 8.23e+00i 3.00e-01 8.62e+00
-2.59e+00 – 8.23e+00i 3.00e-01 8.62e+00
and thus to the open loop flexible mode:
-4.39Є-01 + 8.6ІЄ+00І 5.09Є-02
-4.39e-01 – 8.61e+00i 5.09e-02
This suggests that the robustness of the observed state-feedback controller could be increased by improving the robustness with respect to an uncertainty in the characteristics of this critical bending mode.