684 S. Jiffri et al. D; Dmod Structural damping (phys/ass-mod) x State vector in nonlinear domain E; Emod Structural stiffness (phys/ass-mod) ya;yb;yc Outputs chosen for feedback linearisation f; fmod Applied control forces (phys/ass-mod) z State vector in linearised domain 66.1 Introduction Linear control methods such as pole-placement rely on the assumption that any nonlinearities present in the system are negligible. Such an assumption may be acceptable in weakly nonlinear regimes. However, in situations where behaviour associated with substantial nonlinearity – such as limit cycle oscillations (LCO)–is observed, the approximation of linearity may greatly reduce the effectiveness of active control. Application of linear control methods on a nonlinear system with hardening stiffness is investigated experimentally by Block and Strganac in [1]. It is found that the effectiveness of linear control is limited to situations where the airspeed is not much higher than the linear flutter speed, where the LCO amplitude is small. For airspeeds substantially higher than the linear flutter speed (where LCO amplitudes are higher) the control becomes unpredictable, and its effectiveness limited. Ko et al. apply Feedback Linearisation to a 2-DOF rigid aeroelastic system with torsional nonlinearity [2]. It is shown that using a single control surface leads to local stability, whereas global stability may be achieved using two control surfaces. This work is extended in [3], where the same authors perform a detailed analysis of plunge mode control, and also introduce adaptive feedback linearisation in the two control surface case to account for nonlinearity parameter errors. The latter is implemented experimentally by Platanitis and Strganac [4], with results indicating an improvement when using an additional control surface, but only up to moderately high air velocities. A later publication by Ko et al. [5] examines the case where only a single control surface is employed. In this case, global stability is only guaranteed if the zero-dynamics of the uncontrolled sub-system is stable. The Adaptive control method is later implemented experimentally by Strganac et al. [6], with results suggesting that knowledge of the exact nonlinearity parameters is critical to the performance of feedback linearisation in the absence of adaptive methods, and that the adaptive controller substantially improves the controlled response. It is also observed in [6] that performing feedback linearisation without adaptation in the presence of parameter errors can cause the system to reach non-zero equilibria, rather than the zero equilibrium that is usually sought. Monahemi and Kristic employ adaptive feedback linearisation to suppress wing-rock motion, a phenomenon triggered primarily by aerodynamic nonlinearities [7]. In their work, the role of adaptive control is to update the aerodynamic parameters. Other applications of adaptive feedback linearisation include work by Fossen and Paulsen on the automatic steering of ships [8]. In this paper, Adaptive Feedback Linearisation is applied to a flexible wing with a structural nonlinearity. The latter is introduced to the system by coupling a rigid pylon-engine to the wing via a nonlinear hardening torsional spring. The wing flexibility is modelled using two assumed vibration modes, and is simple in nature (for a detailed model of flexible aircraft, the reader is referred to a publication by Nguyen and Tuzcu [9]). An uncertainty in the parameter describing the nonlinearity is introduced, and the controlled response compared when adaptive feedback linearisation and exact feedback linearisation are applied. 66.2 The Aeroservoelastic Model The governing equation of the aeroservoelastic model takes the usual form given by ARq C. VBCD/ PqC V2CCE q Df; (66.1) where A, D, Eare the inertia, structural damping and structural stiffness matrices respectively, B, Care the aerodynamic damping and aerodynamic stiffness matrices respectively, and ;V are air density and velocity respectively. The vector q contains generalised co-ordinates describing the motion of the system, whereas the vector f contains externally applied generalised forcing terms. In this work, modified aerodynamic strip theory has been used to compute the lift and pitch moment acting on the wing section. An additional unsteady aerodynamic derivative term is included to account for significant unsteady effects.
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