Probabilistic robust linear parameter-varying control of an F-16 aircraft

O PERATIONAL capability at high angles of attack, especially near and at post stall regimes, is critical for next generation fighter aircrafts and uninhabited aerial vehicles [1]. However, significantly large levels of modeling uncertainty are inevitably encountered inflight control design for those regimes. The sources of uncertainty include variations in mass, inertia, and center of gravity positions, uncertainty in the aerodynamic data, etc. [2]. The maneuverability at high angles of attack poses a challenging control problem that requires guaranteeing both robust stability and robust performance in the presence of large parameter variations. Traditional robust control techniques, like H1 and -synthesis, have been proven to be capable of producing robust uncertaintytolerant controllers for next generation aircrafts [2,3]. However, those techniques focus on deterministic worst-case robust analysis and synthesis, which often lead to overly conservative stability bound estimate and high control effort. Moreover, a large number of conventional deterministic problems in robustness analysis and synthesis are shown to be NP-hard. To reduce conservatism and computational complexity, one approach is to shift the meaning of robustness from its usual deterministic sense to a probabilistic one [4]. In contrast to traditional robust control techniques, only a probabilistic solution is given, and a certain risk-level should be accepted. However, such a system may be viewed as being practically robust from an engineering point of view. Algorithms derived in the probabilistic context are based on uncertainty randomization and usually called randomized algorithms, which may be divided into two families: methods based on statistical learning theory [5], and sequential methods based on subgradient iterations [6–8] or ellipsoid iterations [9,10]. The former can deal with nonconvex synthesis problems; however, it resorts to randomized search over the controller parameters to find a candidate solution. On the other hand, the sequential methods are formulated based on convex problems, thus avoiding the controller randomization issue [4]. The probabilistic robust control approach is still in the stage of algorithm development and improvement, and has not been explored in depth for flight control. The number of implementation of probabilistic techniques is therefore rather restricted. In the late 90s, Marrison and Stengel designed a linear quadratic regulator to control the nonlinear longitudinal dynamics of a hypersonic aircraft [11]. Recently, Wang and Stengel designed a robust flight control system for the high-incidence research model problem by combining stochastic robustness with nonlinear dynamic inversion [12]. Their work was based on statistical learning theory, and controllers were searched by using generic algorithms to minimize stochastic robustness cost functions. In our earlier paper, we applied an ellipsoid algorithm to design anH1 controller for a linearized F-16 longitudinal model [13]. Good stability and performance robustness have been achieved at the chosen flight condition. The motivation for this research is twofold. First, the probabilistic control design method for linear time-invariant plants in [13] is generalized to linear parameter-varying (LPV) systems. This generalization is very important because of the relevance of LPV systems to nonlinear systems. TheLPVcontrol synthesis condition is known to be formulated as a convex problemwith a set of parameterdependent linear matrix inequalities (LMIs) [14–16]. Second, the current state of the art does not allow accurate aerodynamicmodeling in the high angle of attack region. Because of its random nature, uncertainty in the aerodynamic data can be characterized using a statistical model, which can be handled effectively by the promising probabilistic robust control approach. Note that the study in this note focuses on the robustness issue with respect to the aerodynamic uncertainty at high angles of attack, and the results would be easily generalized to other parametric uncertainties, such as variations in mass and inertial properties. Because of the convex formulation of LPV control synthesis, the sequential method is more suitable for dealing with uncertainties and designing probabilistic robust LPV controllers. An ellipsoid algorithm with a stopping rule proposed by Oishi [10] is used to determine feasible solutions to LMI synthesis conditions. The paper is organized as follows. In Sec. II, the ellipsoid algorithm is presented, which either gives a probabilistic solution with high confidence or detects that there is no deterministic solution in an approximated sense. Section III first provides a brief overview of robust control problem of an uncertain LPV system, and then discusses the computational issues when the algorithm is applied to the robust LPV control problem. In Sec. IV, a robust LPV controller is designed for an F-16 aircraft with large aerodynamic uncertainty, and the robust performance is tested through nonlinear simulations. Finally, the paper concludes with a summary in Sec. V.