|=||Rotation matrix from a wind frame to the inertial frame|
|=||Control parameter (, , , , )|
|=||System uncertainties and disturbances (, , , , , ,, , )|
|=||Estimate of system uncertainties and disturbances|
|, , ,||=||Moments of inertia|
|=||Control parameter (, , , , , , , ,, , )|
|, , ,||=||Lift, drag, side force, and thrust, respectively|
|, ,||=||Rolling, pitching, and yawing moments|
|, ,||=||The optimal relative positions in the inertial frame|
|=||Mass of the follower aircraft|
|, ,||=||Angular rates in the body frame|
|,||=||The optimal relative positions in the wind frame of the leader aircraft|
|=||Time constant (, , , , , , , , , , , )|
|, ,||=||Airspeed, flight path angle, and heading angle, respectively (, , , , )|
|, ,||=||Resultant airspeed, flight path angle, and heading angle in trailing vortices|
|, ,||=||Wake velocities induced by trailing vortices|
|, ,||=||Estimates of wake velocities|
|, ,||=||Position coordinates (, , , )|
|, ,||=||Position errors in the inertial frame|
|, ,||=||Angle of attack, sideslip angle, and bank angle, respectively|
|, ,||=||Vortex-induced lift, drag, and side force, respectively|
|, ,||=||Vortex-induced moments|
|, ,||=||Aileron, elevator, and rudder|
|=||Damping ratio (, , , , , , , )|
|=||States of disturbance observer (, , , , , , , , , , , )|
|=||Auxiliary signals (, , , , )|
|=||Natural frequencies (, , , , , , , )|
Close formation flight is inspired by migratory birds who adopt a “V-shape” formation flight when migrating from one habitat to another Lissaman1970Science ; Weimerskirch2001Nature ; Portugal2014Nature . In close formation, a follower aircraft, holding a close relative position to a leader aircraft, flies in the upwash wake region of the trailing vortices induced by the leader aircraft as shown in Figure 1, by which the follower aircraft reduces its drag and thus saves fuels. Drag reduction of close formation flight has been demonstrated by simulations Halaas2014SciTech ; Kent2015JGCD ; Zhang2017JA , observed by wind tunnel experiments Bangash2006JA ; Cho2014JMST , and confirmed by flight tests Ray2002AFMCE ; Pahle2012AFMC ; Bieniawski2014ASM ; Hanson2018AFMC .
Despite its benefits, close formation flight is challenging in terms of the accuracy and robustness requirement for guidance and control Zhang2017JA . The position control accuracy must be guaranteed within the consideration of system uncertainties and formation-related aerodynamic disturbances. Yet, different control algorithms have been proposed for close formation flight. Most of them are focused on level and straight flight with constant speeds Schumacher2000ACC ; Pachter2001JGCD ; Dogan2005JGCD ; Almeida2015GNC . Two different linear strategies were applied, namely formation holding control and formation tracking control. Both of them are limited. Formation-holding control assumes a follower aircraft is initially well-trimmed at its optimal position in close formation, such as the proportional-integral (PI) formation control Pachter2001JGCD , the close formation control by the linear-quadratic regulator (LQR) Dogan2005JGCD , and the linear model predictive control (MPC)-based control Almeida2015GNC . Linear formation-tracking control doesn’t require the follower aircraft to be initially located at its desired position in close formation Binetti2003JGCD ; Lavretsky2003GNC ; Chichka2006JGCD ; Zhang2016GNC ; Zhang2017GNC ; Zhang2018AESCTE , but they are not guaranteed to address complex nonlinear aircraft dynamics at dynamic flight operation. Additionally, linear control methods will experience dramatic performance degradation or even fail to stabilize the system, when being applied to nonlinear systems. Robust nonlinear formation control is, therefore, more preferable to accommodate close formation flight at dynamic operation.
Nonlinear close formation control is challenging. Contrary to linear cases, model uncertainties and aerodynamic disturbances are less predictable and harder to be described at nonlinear scenarios, making close formation control more difficult. Early robust nonlinear close formation control was investigated using sliding model control Singh2000IJRNC or high order sliding mode (HOSM) control Galzi2007ACC . Both of the two methods only focus on outer-loop design, and they require the vortex-upper bounds of induced forces or their derivatives to satisfy certain limits to guarantee the stability. The nonlinear robust formation design including both inner-loop and outer-loop control was reported in Brodecki2015JGCD using an incremental nonlinear dynamic inversion (INDI) method, but this method is not robust to model uncertainties and external disturbances. Therefore, present nonlinear methods either rely on unknown model information to ensure both stability and robustness like the sliding mode control Galzi2007ACC , or not robust to model uncertainties and external disturbances at general dynamic operations, such as the INDI-based control Brodecki2015JGCD . Therefore, it is still an open issue for nonlinear robust close formation control with certain performance guaranty only using available model information.
In this paper, we investigate the robust nonlinear control problem for close formation flight at dynamic operation. The control design is presented under a leader-following architecture. The fundamental objective is to secure highly precise position control for close formation flight at dynamic flight operation design with the consideration of system uncertainties and aerodynamic impact caused by trailing vortices of leader aircraft a robust nonlinear controller. Control robustness will be one of the critical concerns which significantly affects the possible accuracy for close formation flight as it is subject to system uncertainties and aerodynamic disturbances. A robust nonlinear formation controller, which consists of baseline controllers and disturbance observers, is proposed in this paper. The baseline controllers are designed based on a command filtered backstepping technique to stabilize the nominal nonlinear dynamics of an aircraft in close formation Farrell2005JGCD ; Farrell2009TAC , whereas the disturbance observers could estimate and compensate for system uncertainties and formation-related aerodynamic disturbances by purely observing system inputs and available states. In the proposed design, the follower aircraft is required to track its dynamic optimal relative position to a leader aircraft in the inertial frame under different flight maneuvers. Both the inner-loop and outer-loop control will be studied in this paper, which makes the formation control design more reliably but also more difficult. The assumption on a well-designed inner-loop controller in Zhang2018TIE is, therefore, removed in this paper. The proposed design is capable of achieving highly accurate and efficiently robust control performance without using any gradient or boundary information of formation aerodynamic disturbances. Position tracking errors will be ultimately bounded. The final boundaries could be regulated by choosing different control parameters.
Some preliminaries are provided, which will be used for the design and analysis in the sequel.
Definition 1 (Definition 4.6 Khalil2002Book ).
A system is uniformly ultimately bounded if there are positive constants and , there exists for any , such that
Let be a bounded signal whose derivative , namely and . Assume is the estimation of through a first-order filter as shown in (2).
where is a time constant. Define as the estimation error. If ,
are globally bounded by ;
if , there exist any small positive constant and time such that for all , where ;
if , .
Iii Formulation of reference trajectories at dynamic operation
In this section, a motion planner is designed for follower aircraft at close formation. According to Zhang2017JA , the optimal relative position in close formation is static in the wind frame of the leader aircraft. Assume is the static optimal relative position in the wind frame of the leader aircraft, where ranges from to and is around with denoting the wing span. When flying at close formation, the reference position of a follower aircraft in the inertial frame is
where , , and are position coordinates of the leader aircraft in the inertial frame, and where , , and are the bank, flight path, and heading angles of the leader aircraft. Differentiating (3) yields
where , , and are the airspeed, flight path angle, and heading angle of leader aircraft, respectively. At dynamic operation, , , and are time-varying, but their derivatives cannot be accurately computed. Hence, in the design, we introduce a command filter (5) to get the command signals and (). Let be a smooth signal, so the command filter is
where is the natural frequency and is the damping ratio. Let and . If and , Lemma 2 exists for any bounded signal .
The estimator (5) is input-to-state stable with respect to . If both and are bounded, is uniformly and ultimately bounded, and the following inequality exists for .
where and are the maximum and minimum eigenvalues of a matrix, respectively,
are the maximum and minimum eigenvalues of a matrix, respectively,is positive definite such that . Furthermore, there exist
where is an order of magnitude notation Khalil2002Book .
In real flight, , , and and their derivatives are all bounded, so (5) is a valid choice to obtain and (). In addition, Lemma 2 indicates that the command signals and () could be ensured to be arbitrarily close to their corresponding desired values and by choosing proper command filter parameters. Note that is needed to avoid the peaking phenomenon (Page 613, Khalil2002Book ). If , , the signal will transiently peak to before it exponentially decays, resulting in the so-called peaking phenomenon due to . Without loss of generality, the following assumption is introduced.
The attitude signals , , and and their derivatives are all bounded.
In light of (5), the command position for a follower aircraft in close formation is , , and , and accordingly,
Iv Robust nonlinear formation control design
The proposed design in this section can be easily extended to the case with more than three aircraft, even though it is discussed under the leader-follower architecture with two aircraft. In the proposed design, command filtered backstepping technique is employed, which avoids the analytic calculation of time derivatives of intermediate virtual inputs Farrell2005JGCD ; Sonneveldt2007JGCD ; Farrell2009TAC ; Sonneveldt2009JGCD . As shown in Figure 2, the entire design consists of two major loops: an outer loop for formation position control and an inner loop for attitude control. The outer-loop control allows a follower aircraft to track the planned motion by (6), and generates command thrust , desired bank angle , and desired angle of attack . The inner-loop control stabilizes follower aircraft’s attitudes to their desired values and from the outer-loop control, while holding zero sideslip angle .
iv.1 Outer-loop formation position control
Let and be the nominal values of the drag and lift , respectively. They are obtained by either available aerodynamic data or certain analytical models Morelli1998ACC . The sideslip angle is negligibly small, as it is always stabilized to be zero. Accordingly, the side force is small and taken as a model uncertainty. The outer-loop dynamics used for control design are
where , , and are follower position coordinates in the inertial frame, is the airspeed, and are the flight path and heading angles, is the thrust, , , and are induced wake velocities, and , , and are the augmentation of system uncertainties and disturbances.
where , , and are the wake velocity derivatives denoted in the wind frame of follower aircraft, , , and are the vortex-induced forces. According to Zhang2017JA , , , and are bounded, and have much slower dynamics in comparison with aircraft speed and attitudes, so their derivatives are relatively small. Furthermore, the following assumption is introduced.
Induced wake velocities , , and are all bounded, and furthermore, they are piecewise constant, namely , , and .
The following nonlinear disturbance observer is employed.
where , , , , where , , and are estimates of , , and , respectively. It is chosen that . Let , , and . Under Assumption 2, one has
Let , , and . Transform , , and into a new frame.
where . The desired velocity and flight path angle are shown in (13).
where are control parameters, and . The desired signals and are passed through a command filter to obtain , , and their rates. Hence,
Let and . Note that where is the lift at and is the lift derivative with respect to the angle of attack. According to (7), one has
where , , and are intermediate control inputs, is the estimation of by passing through a 2nd-order filter similar to (5). Two more uncertainty terms and are included in in (15), where and . Hence, in (15) is re-defined to be . Note that is a smooth signal with bounded derivatives. According to Lemma 2, and its derivative are bounded, so and its derivative are also bounded.
The uncertainties and disturbances , , , and their derivatives are bounded.
where , , and are estimates of , , and , respectively, and , , and are
Shown in Figure 3 is the command filter and auxiliary system for speed control. If one chooses , , and , there exists
Based on (19), the nonlinear disturbance observer is
where is a positive definite constant matrix, , , , and . The uncertainty and disturbance estimates , , and need to be fed back to the estimator for the next estimation updates. Combining (17) and (20), one is able to get , , and . Hence,
The entire outer-loop control structure is illustrated in Figure 4. The following assumption is introduced for , , and for the stability analysis.
, , and have slow dynamics, and furthermore,