By Simon McBeath
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Additional resources for Aerodinámica del automóvil de competición
This is a Kalman - like rank condition. Since one can use only sign constrained inputs, this imposes an additional condition on H. A suﬃcient condition is that if H has an even number of eigenvalues with zero real parts, then the zero dynamics is controllable with nonnegative inputs. More results on controllability with nonnegative inputs can be found in [11, 13]. This result is extended for our application as follows. Consider the MISO system with B ∈ Rn×m and ys = Cx. Also consider the case, when there is a direction p ∈ ImB such that the system is left and right invertible cor¯ one has the responding to the direction p.
Arndt z (m) 20 Basic AMP 15 AMP 20 10 0 −10 0 1 2 3 4 5 6 7 1 2 3 4 time (s) 5 6 7 w (m/s) 10 5 0 −5 −10 0 Fig. 5. Simulated trajectory of high speed supercavtating vehicle unity actuator model. Figure 7 shows the eﬀect of including a 30 Hz, ﬁrst order actuator model. The performance speciﬁcations are to track θ reference commands and reduce limit cycle oscillations. The reference tracking properties received the higher priority as compared with the damping the oscillatory behavior. The following controller gains were selected: −α ¯ 110 = −40000; −α ¯ 111 = −400; −α ¯ 220 = −90000; −α ¯ 221 = −600 (44) With which the resulting eigenvalues are −300; −300; −200; −200.
The following approach is taken. Since only one delay time is present in the switching condition, it is possible to discretize the system with extended state space by including the delayed state variable. Since feedback linearization has been already applied, it is possible to use a backward diﬀerence scheme deﬁned for LTI systems that preserves the geometry needed to analyze the zero dynamics. The resulting discrete time state equations are: x(t + 1) = Ad x(t) + Bd v(t), where 1 0 Ad = 0 0 1 T 1 0 0 0 0 0 0 0 1T 0 1 0 0 0 0 0 , 0 0 0 0 ys = Cd x(t) (33) β21 T β22 T 0 Bd = 0 , β41 T β42 T 0 0 Cd = [1, 0, v, 0, −1] (34) where T denoted the sample time.
Aerodinámica del automóvil de competición by Simon McBeath