66 Adaptive Feedback Linearisation and Control of a Flexible Aircraft Wing 693 Since the entire three dimensional nonlinear system is linearised to give a three dimensional linear system with the aid of multiple inputs and outputs, the complete dynamics of the original system is preserved in the linearised system (thus, the linearised system will not have Zero Dynamics in this case). 66.5 Numerical Simulation (Part 2) The numerical simulation from Sect. 66.3 is now continued. Closed-loop control is applied to the nonlinear model via the two control surfaces and the torsional actuator, to provide linearising feedback and thereby perform pole-placement. The feedback is computed such that the linearised system consists of the three uncoupled SDOF sub-systems referred to in the above section. The natural frequencies and damping ratios specified in the pole-placement of the three systems is shown in Table 66.3. The system is simulated with the same initial conditions as before, but with feedback linearisation implemented from the beginning. The resulting controlled response of the assumed-modes co-ordinates is shown in Fig. 66.10. The top three sub-plots show the displacements of the generalised co-ordinates, whereas the bottom three show the respective velocities. It is immediately evident that the LCO of the uncontrolled nonlinear system has been eliminated. Furthermore, as shown in Fig. 66.11, a horizontal zoom of the first second of response shows that the intended change in the natural frequencies has been achieved. The corresponding physical displacement plots are shown in Fig. 66.12, and the actual inputs required for the above feedback linearisation is shown in Fig. 66.13. It can be seen that the intended objectives of the feedback linearisation have been fulfilled with achievable inputs from the two control surfaces and torsional actuator. Table 66.3 Natural frequencies and damping ratios specified in Pole-placement of linearised system q1 0.93Hz 0.01 q2 4.95Hz 0.01 qpe 2.9Hz 0.01 0 20 40 60 -1 -0.5 0 0.5 1 x 10-4 Time (s) q1 q2 0 20 40 60 -1 -0.5 0 0.5 1 x 10-3 Time (s) q1 dot q2dot 0 5 10 15 -2 -1 0 1 2 x 10-3 Time (s) 0 5 10 15 -0.1 -0.05 0 0.05 0.1 Time (s) 0 5 10 15 20 -0.1 -0.05 0 0.05 0.1 Time (s) qpe 0 5 10 15 20 -1 -0.5 0 0.5 1 Time (s) qpedot Fig. 66.10 Feedback-linearised response at 80 ms 1 (assumed-modes co-ordinates)
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