Robust MRAC augmentation of flight control laws for center of gravity adaptation

When an aircraft is flying and burning fuel the center of gravity (c.g.) of the aircraft shifts slowly. The c.g. can also be shifted abruptly when e.g. a fighter aircraft releases a weapon. The shift in c.g. is difficult to measure or estimate so the flight control systems need to be robustly designed to cope with this variation. However for fighter aircrafts with high manoeuvrability there is room for improvements. In this project we investigate if the use of adaptive control law augmentation can be used to better cope with the change in c.g. We augment a baseline controller with a robust Model Reference Adaptive Control (MRAC) design and analyse its benefits and possible issues. 1 Aircraft model and baseline controller The dynamics that we will consider in this report is a linearized version of the pitch dynamics of the ADMIRE aircraft [Forssell and Nilsson, 2005] on the form ẋ = Ax+Bu, y = Cx (1) where x = [ α q ]T and α is the angle of attack and q is the pitch rate of the aircraft, see Figure 1, and u is the elevator control surface deflection. The matrices A and B vary when the c.g. shifts from Figure 1: Definition of angles for aircraft control its most forward position to its most aft (backward) position. With the c.g. in the most forward position the matrices are A = [ −1.453 0.9672 5.181 −1.639 ] , B = [ 0.4467 34.79 ] (2) and when the c.g. is in the most aft position A = [ −1.45 0.9673 15.08 −1.414 ] , B = [ 0.4461 31.77 ] (3) From this we can see that the force equation (first row of the matrices) is almost unaffected by the c.g. shift while the moment equation is largely affected by the shift in c.g. To stress the adaptive controller as much as possible we have designed the baseline controller for the most forward c.g. case and then simulate the total system with the model of the most aft c.g. case. The baseline controller consist of an LQ feedback term, a static feed forward term to get a static gain of one between the reference and the output (angle of attack) and finally a integral part which 1 ar X iv :1 60 4. 01 88 2v 1 [ cs .S Y ] 7 A pr 2 01 6 integrates the error between the output and the nominal closed loop response (without the integral part). The baseline control signal is thus ubl = −Kx+ Fr + ∫ (y − yref )dt where y = α and yref = C(sI − A + BK)−1BFr. The baseline controller is designed using the matrices A and B from (2) but the B matrix is simplified by setting the element in the force equation to zero, i.e., assuming that the control surface deflection do not generate any lift force but only moment. This approximation is not necessary at this stage but will have some nice implications in the adaptive design. In Figure 2 the response of the closed loop system with the nominal controller and different c.g. positions is shown. We can see that the response is good when the c.g. is at its nominal position (the blue line) but as the c.g. is moving backwards (green and red lines) there is a large overshoot. It is this overshoot that we want to minimize with an adaptive augmentation without destroying the nominal performance of the baseline controller. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 From: ref To: AoA Step Response Time (seconds) A m pl itu de Nominal c.g. 50% aft. c.g. 100% aft. c.g. Figure 2: Step response of the closed loop system with nominal controller for different c.g positions 2 Robust MRAC design In Model Reference Adaptive Control (MRAC) one compare the output (y) of the closed loop system with that from a reference model (ym). Then the controller parameters are updated such that the closed loop system response is as close as possible to that of the reference system. The parameter update can be done in several different ways, e.g., by using the MIT-rule or by using Lyapunov stability theory. In this project we have chosen to use Lyapunov stability theory to derive the update laws for the controller parameters. This is mainly due to the theoretical stability guarantees that comes with the method. In this report we will only briefly describe the Lyapunov design process. For more information on the theoretical background of MRAC and the MIT and Lypunov update rules we refer the reader to the books of Åström and Wittenmark [2008], Ioannou and Sun [2012], Lavretsky and Wise [2013]. In the MRAC design technique that we have adopted the uncertain system is modeled as ẋ = Ax+BΛ(u+ θφ(x)) (4) where A is an unknown matrix, Λ is an unknown diagonal matrix and B is known. The vector θ is the unknown coefficients of the general nonlinear function θφ(x) where φ(x) is a set basis functions. The aim of the adaptive controller is to have the system (4) follow a reference model ẋ = Amx+Bmr (5)