Example: The X-29
Considerable design effort has been devoted to the design of the flight control system for the X-29 aircraft, see [12]and [41]. One of the design criteria was that the phase margin should be greater than 45 for all flight conditions . at one flight condition the model has the following non-minimum phase component
Since z= 4.33p it follows from Equation (40) that the achievable phase margins for
It is interesting to note that many design methods were used in a futile attempt to reach the design goal A simple
calculation of the type given in this section would have given much insight.
Example: Klein�s Unridable Bicycle
An interesting bicycle with rear wheel steering which is impossible to ride was designed by professor Klein in
lllinois, see[33]. The theory presented in this paper is well suited to explain why it is impossible to ride this
bicycle. The transfer function from steering angle to till angle is given by
Where m is the total mass of the bicycle and the rider, j the moment of inertia for tilt with respect to the contact line of the wheels and the ground the height of the center of mass from the ground, a the vertical distance from the center of mass to the contact point of the front V the forward velocity and g the acceleration of gravity. The system has a RHP pole at s=p= caused by the pendulum effect. Because of the rear wheel steering the system also has a RHP zero at
The reason why the bicycle is impossible to ride is thus that the system has a right half plane and a right half plane zero that are too close together. Klein has verified this experimentally by making a bicycle where the ratio z/p is larger. This bicycle is indeed possible to ride.
So far we have only discussed the case z>p . when the unstable zero is slower than the unstable pole the crossover frequency inequality (29) cannot be satisfied unless
4.5 A Pole in the Right Half Plane and Time Delay
Consider a system with one pole in the right half plane and a time delay T the non-minimum phase part of the transfer function is thus
Example: Pole balancing
To illustrate the results we can consider balancing of an inverted pendulum. A pendulum of length l has a right half plane pole g/l assuming that the neural lag of a human is 0.07s. the inequality (43)gives g/l0.07 less than0.326,hence l>0.45. the calculation thus indicate that a human with a lag of 0.07 s should be able to balance a pendulum whose length is 0.5m. to balance a pendulum whose length is 0.1 m the time delay must be less than 0.03 s. Pendulum
balancing has also been done using video cameras as angle sensors. The limited video rate imposes strong
limitations on what can that can balanced with
4.6 Other criteria
The phase margin is a crude indicator of the stability margin. By carrying out detailed designs the results can be refined. This is done in [1] which gives results for designs with
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