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Unformatted text preview: 36 CONTROL OF A HIGH-SPEED VEHICLE 134 36 Control of a High-Speed Vehicle An instability in certain aircraft and in some high-speed marine vehicles is characterized by a pair of complex right-half plane poles, and is due to inadequate aerodynamic stability. It is particularly pronounced in highly maneuverable craft, where open-loop stability in the physical design has been intentionally traded off against agility. In this problem, you will use the Nyquist criterion to bring such a vehicle under control and to achieve specific performance and robustness properties. 1. The transfer function taking the elevator control surface command into the pitch of the vehicle is given as y ( s ) 2( s + 2) P ( s ) = = u ( s ) s 2 − . 2 s + 16 Confirm that this is an unstable system by both stating the poles and by plotting the impulse response. 2. Plot the complex loci for this plant. This means plotting the real versus the imaginary parts of P ( s ), s = jω , for ω going from zero to infinity. Then plot this same curve again, negating the imaginary part, and you then have covered the range of ω from [ −∞ , ∞ ]. Use logarithmically-spaced frequencies. 3. Applying the Nyquist criterion to this plot, can the vehicle be stabilized with unity feedback gain, i.e., C ( s ) = 1? If so, comment on whether it is a useful feedback system. 4. We need to achieve through feedback a) integral action, b) a phase margin of 30-40 degrees, and c) a gain margin of at least two. To do this, design a controller C ( s ) with the following properties: • a real pole just to the left of the origin, corresponding with an integrator (we want it to be definitely not in the RHP); • a stable complex zero pair that approximately mirrors the unstable plant poles across the imaginary axis, but possibly with a higher damping ratio; • two additional stable, real poles so that the compensator has a high-frequency roll-off behavior....
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