Constrained Linear Quadratic Tracker for Optimal Performance Flight

Abstract

M ODERN autonomous vehicles are able not only to follow certain low-level directives, such as airspeed, altitude, and heading, but also to track computed optimal trajectories in the sense to minimize fuel consumption and time of arrival at a certain waypoint while avoiding collisions with known and unknown obstacles, the terrain and other aerial vehicles. Substantial effort has beenmade recently to develop several techniques to compute optimal constrained trajectories for manned [1] and unmanned aircraft [2,3]. A common characteristic of these methods is their strong dependency on the maneuverability properties of the vehicle under consideration. Variables such as flight-path angle, turning rate, and turning radius are considered as constraints in the related optimization problems. Thus, larger values of flight-path angle and turning rate and a reduced radius of turn are desirable, because these improvements would allow for the operation of the vehicle in a wider set of scenarios. However, the inclusion of performance and maneuverability constraints is often accomplished in terms of maximum and minimum bounds without accounting for the fact that, in an aerial vehicle, optimal performance is always related to airspeed or Mach number [4]. For example, it is unlikely to simultaneously achieve maximum airspeed and maximum turning rate. The aircraft flight envelope, i.e. the combination of all achievable steady-state conditions, is more complex than a simple rectangle formed by minimum and maximum admissible values. Once the flight envelope is properly defined for operational purposes, the next task is the proposal and synthesis of a controller concerned with the defined limitations. This controller function, commonly referred to as an envelope protection system, is now widespread in the aeronautics industry [5] and remains a current topic of research. Unnikrishnan and Prasad [6] presented a protection system inwhich the operational limitations are treated as obstacles to be avoided. Falkena et al. [7] investigated several approaches to the problem of envelope protection, including a nonlinear constrained control law based on model predictive control (MPC). One of the challenges to designing envelope protection systems is the need to predict not only the future steady-state level of the constrained variable but also its transient characteristics, especially the peak value. Thus, a natural approach to this problem takes into consideration constrained control techniques, amethodology that has been validated by numerous studies [8–10]. The key mathematical element of these stable constrained controlmethods is the presence of an invariant set [11]withinwhich the state and input are guaranteed to lie, in accordance with the system constraints, indefinitely. Thus, strategies based on the invariant setmembership of the statevector are adequate to handle the envelope protection problem. Based on the aforementioned problems and current state-of-art solutions, this Note presents a constrained control technique called the constrained linear quadratic tracker (CLQT). The proposed method is a simplified variant of the MPC method known as the feasible target tracking MPC [10]. In addition, a new method to represent theflight envelope is proposed, based on linear inequalities. The denominated polyhedral flight envelope is constructed from the computed solutions of the nonlinear free-body equations with the advantage of providing achievable speed schedules for maximum performance maneuvers, such as maximum climbing flight-path angle, maximum turn rate, or a combination of both. Simulation results of the complete system composed of the polyhedral flight envelope representation and CLQT, using the Acaua test vehicle model (Fig. 1), are obtained in calm and turbulent air. The results demonstrate suitable tracking performance and protection against stall, even in a severely turbulent environment, with a relatively low computational execution time.