ROBUST PERFORMANCE: POLE LOCATION (P2)
Robust stability inside a truncated sector is studied. The minimal degree of stability (resp. the minimal damping ratio) is chosen as 0.3 (resp. 0.4). The nominal degree of stability (resp. the nominal damping ratio) is 0.49 (resp. 0.62).
Figure 6.3. Missile autopilot – robust stability inside a truncated sector (P2) – the real /і upper bound by Zadeli and Desoer is represented in dashed line and the real lower bound by Dailey in solid line.
We proceed in the same way as in the previous subsection. The same 5 methods are used. Here again, all these methods give nearly the same result (around 0.12) at ш = 0. We then focus on the real // upper bound
by Zadeh and Desoer (dashed line – see Figure 6.3) and the real lower bound by Dailey (solid line). Note here again the good accuracy of the p interval at nearly all frequencies, and the discontinuity of the p bounds by Zadeh and Desoer and by Dailey at и = 0: p w 0.12 at и = 0, while p « 0.06 at very low frequencies. The real s. s.v. is thus discontinuous at the zero frequency.
Missile autopilot – robust stability inside a truncated sector (P2) – the
classical complex ц upper bound is represented in solid line, the real и upper bound by Jones in dashed line, the mixed ц upper bound by Fan et al in dash-dotted line and the real fi upper bound by Zadeh and Desoer in dotted line.
The maximal value of the upper bound by Zadeh and Desoer is 0.25 at u) = 13.6 rad/s, while the maximal value of the lower bound by Dailey is 0.24 at з = 1 3.5 rad/s. The accuracy of the estimate of the robustness margin is thus very good (less than 5 %). The corresponding uncertainties in the stability derivatives are 5/0.25 = 20%. The Hoo missile autopilot consequently presents good robust performance properties in
the presence of uncertainties in the aerodynamic model.
When analyzing robust stability inside the left half plane, two peaks were obtained at ш = 0 (the real ц was discontinuous at this frequency) and at medium frequencies ( a a 1 4.4 sa4/s( . In the same way, when analyzing robust stability inside a truncated sector, two peaks are here again obtained at u> = 0 and at medium frequencies (со и 13.6 rad/s).
As a final point, Figure 6.4 presents the results obtained with the four fj, upper bounds (classical complex /r upper bound in solid line, real ц upper bound by Jones in dashed line, mixed ц upper bound by Fan et al in dash-dotted line, real ц upper bound by Zadeh and Desoer in dotted line). The same comments can be done as for Figure 6.2.