Heavyweight airdrop flight control design using feedback linearization and adaptive sliding mode

Abstract

This paper investigates the problem of designing a novel adaptive sliding-mode controller for heavyweight airdrop operations. The design objective is to guarantee asymptotic tracking performance of the aircraft states, in the presence of bounded nonlinear uncertainties without prior knowledge of the bounds. On the basis of feedback linearization of the aircraft–cargo model, a sliding-mode control method with projection-based adaptive function approximation is proposed. This method uses an adaptation strategy to achieve robustness against model uncertainties, and a knowledge of the bounds on the complex uncertainties is not required. Notably, the adaptation law with projection can bound the estimated function, and this avoids singularity of the control signal. Simulations are conducted under the condition that one transport aircraft performs a maximum load airdrop mission at a height of 25 m, using single-row single-platform mode. The results verify the good properties of the control method, which can meet the airdrop mission performance indexes well in the presence of ±20% aerodynamic data uncertainty and 20% actuator fault.

Keywords

Nonlinear system feedback linearization uncertainty flight control sliding-mode control adaptive control

Introduction

The heavyweight airdrop is an essential capability of a large transport aircraft, and it is critical to the success of many military tasks, such as precision delivery of heavyweight equipment and supplies (Desabrais et al., 2012; Geisbauer et al., 2011; Jann, 2011). During standard airdrop operations, cargoes are released at altitudes of 15 m to 450 m at aircraft velocities between 62 m s⁻¹ and 77 m s⁻¹ to avoid enemy radar detection and the effects of anti-aircraft artillery (Henry et al., 2009; Pang et al., 2013). Also, the low-level and low-speed flight characteristics can effectively reduce collateral damage to the cargo. To perform tasks perfectly with accurate allocation of the payloads and also to guarantee flight safety, highly steady aircraft dynamics are required. However, movement of the payload exerts large disturbances on the aircraft, thus leading to considerable deviation of the aircraft’s dynamics from the trim position (Chen and Shi, 2009; Cuthbert et al., 2005; Feng et al., 2011; Li et al., 2013; Liu et al., 2015a, 2015b; Raissi et al., 2008; Yang and Lu, 2012; Zhang et al., 2014). To hold the flight state a pitch-down control action is required, followed by an abrupt change in the direction of the control force. This manipulation is quite sophisticated, and does not leave a large margin for operational error (Raissi et al., 2008). Therefore, the design of an aircraft controller for the heavyweight airdrop mode is necessary; and this is also a challenging task due to the occurrence of large and sudden disturbances, strong coupling between the cargo and aircraft dynamics and multiple uncertainties (Feng et al., 2011; Li et al., 2013; Liu et al., 2015a, 2015b; Yang and Lu, 2012).

Over recent years, some meaningful achievements have been reported in the development of airdrop mode advanced aircraft controllers. Several control methods using a linear model at a given operating point have been seen in the literature, including L₁ adaptive control (Liu et al., 2015b), model reference adaptive control (Tang et al., 2011) and active disturbance rejection control (Yang et al., 2010, 2011; Zhang et al., 2009a). Although these methods can improve aspects of the performance of the system, one problem is that controllers with this design may have unsatisfactory performance as the cargo becomes increasingly heavy. In such an event, the aircraft dynamics can deviate far from the operating point during the cargo extraction phase. Feng et al. (2011) linearized the aircraft model as a train of operating points throughout the airdrop process, but this is rather tedious work and cannot fundamentally solve the problem mentioned above. To further improve the performance of systems with strong nonlinearities, many nonlinear control approaches have been developed. The theoretically established feedback linearization method is the one that has been most widely applied (da Costa et al., 2003; Sieberling et al., 2010; Rehman et al., 2013; Choi and Bang, 2014; Le et al., 2014).

Feedback linearization (Slotine and Li, 1991; Kotta et al., 2013) is a nonlinear design approach that can explicitly handle nonlinear systems, and it has been widely used in reentry flight control (Da Costa et al., 2003), UAV control (Choi and Bang, 2014; Sieberling et al., 2010), hypersonic flight vehicle control (Rehman et al., 2013), motion control for cranes (Le et al., 2014) and so on. By using nonlinear feedback and exact state transformations rather than linear approximations, the nonlinear system is transformed into a constant linear system. However, to perform perfect linearization, accurate knowledge of the plant dynamics should be available. This is not the case with a design for airdrop flight control, because it is very difficult to precisely know and model the complex aerodynamic characteristics (Chen and Shi, 2009; Liu et al., 2015a). Moreover, aerodynamic data obtained from wind tunnel tests always contains a certain degree of uncertainty. It is well known that sliding-mode control (SMC) is an efficient approach for handling model deficiencies. On the basis of feedback linearization of the system, Zhang and Shi, (2009b) and Li et al. (2013) designed a linear sliding-mode controller to control an aircraft’s speed and pitch angle. Yang and Lu (2012) designed a backstepping sliding-mode controller for stabilizing an aircraft’s altitude that is able to solve the cargo airdrop mismatched uncertain control problem. Liu et al. (2015a) proposed an iterative quasi-sliding-mode control strategy for the control of airdrop mode pitch attitude, and this strategy can guarantee global robustness of the sliding motion by introducing a reference approaching function at the first level sliding mode, and, further, can achieve high-precision control performance through the design of the second level iterative integral sliding mode. In spite of their obvious conceptual appeal (Li et al., 2013; Liu et al., 2015a; Yang and Lu, 2012; Zhang et al., 2009b), these methods require the upper bounds on the uncertainties to specify the control gains to satisfy the requirements of stability and robustness. However, the complex uncertainties, composed of aircraft–cargo dynamics coupled with aerodynamic data perturbation, are always difficult to achieve in advance. Thus, the control gains need to be set large enough, usually very conservatively, which might further lead to severe chattering and might damage actuators and systems (Boiko, 2011; Boiko et al., 2007; Levant, 2010).

One of the widely applied methods to alleviate chattering is the use of a boundary layer, thus leading to the concept of quasi-sliding-mode control (Slotine and Sastry, 1983). Another efficient method for reducing chattering is the use of higher order sliding-mode control (Levant, 1993; Laghrouche et al., 2007). However, in both of these control methods, the selection of switching gains still relies on knowledge of the bounds on the uncertainties. As the objective is the non-requirement of the bounds on the uncertainties, the concept of combining SMC with adaptation algorithms has attracted the attention of many scholars. Huang et al. (2008) proposed an adaptive SMC method for a class of uncertain nonlinear systems. The method adapts switching control gains directly, depending on the tracking error of the sliding function to counteract uncertainties, and introduces a boundary layer neighbouring the sliding surface for real applications. This means that control accuracy has to be sacrificed due to the introduction of the boundary layer. Lee and Utkin (2007) applied an equivalent control concept to adjust the switching control gains. Yet their method still requires knowledge of the bounds of the uncertainties, and the use of a low-pass filter, which might lead to a time delay. In Plestan et al. (2010, 2013), a gain-adaptation algorithm that depends on the distance of the sliding surface to a constant is proposed, and the knowledge of the bounds of the uncertainties is not required. However, they can only guarantee that the sliding surface converges in the neighbourhood of the origin.

The main motivations for the current work are to propose a systematic and simple adaptive SMC design for the heavyweight airdrop mode, in the presence of bounded nonlinear uncertainties without prior knowledge of the bounds. The contributions of the current paper are: (1) the multivariate cross-coupling aircraft–cargo model is decoupled via feedback linearization, and the difficulties of designing the control system for the airdrop mode are greatly reduced; (2) unlike previous works, the complex nonlinear uncertainty of the aircraft–cargo system is factorized into a known matrix and an uncertainty function, and we use projection-based adaptive algorithms to approximate the function. The sliding manifolds as well as the flight state tracking errors asymptotically converge to zeroes with non-requirement of the bounds of the uncertainty; and (3) the adaptation law formed using the projection operator (Natarajan and Bentsman, 2014; Pomet and Praly, 1992) can bound the estimated function, and this avoids a singularity of the control signal.

The current paper is organized as follows. The aircraft–cargo coupled model is presented in the following section; then the model is decoupled via the input–output feedback linearization technique. The adaptive SMC law for the airdrop mode is derived and the stability performance is discussed. A simulation analysis is presented and, finally, some conclusions are drawn.

Aircraft–cargo coupled model

During the airdrop process, the cargo moves along the rail on the cargo deck with the help of the extraction system. For convenience, the cargo is considered as a particle, and the rail is assumed to coincide with the aircraft’s longitudinal body axis $o x_{b}$ , as shown in Figure 1. This means that the cargo affects the longitudinal dynamics of the aircraft but does not influence its lateral dynamics. The disturbances that the cargo exerts on the aircraft include $F_{c x}$ , $F_{c z}$ and $M_{c}$ , which are the friction force, contact force and pitch moment, respectively. The aerodynamic forces acting on the aircraft are the drag force D and lift force L. Furthermore, the pitch aerodynamic moment and the engine thrust are represented by $M_{y}$ and T, respectively. The pitch status of the aircraft is characterized by V, $α$ , $γ$ and $θ$ , which are the airspeed, angle of attack (AOA), climb angle and pitch angle, respectively.

Figure 1.

Graphical representation of the airdrop process and forces analysis of the aircraft.

Using Newton’s second law, the dynamics of an aircraft with cargo moving inside (Liu et al., 2015a) can be derived as follows

\dot{V} = (T \cos α - D - m_{b} g \sin γ - F_{c x}) / m_{b}

(1)

\overset{\cdot}{γ} = (T \sin α + L - m_{b} g \cos γ - F_{c z}) / m_{b} V

(2)

\overset{\cdot}{q} = (M_{y} + M_{c}) / I_{y}

(3)

\overset{\cdot}{θ} = q

(4)

where $m_{b}$ is the mass of the aircraft; q is the pitch rate; and $I_{y}$ is the pitch moment of inertia.

The pitch aerodynamic moment is obtained as

M_{y} = \bar{q} S c_{A} [C_{m 0} + C_{m α} (α - α_{0}) + C_{mq} \frac{q c_{A}}{2 V} + C_{m δ_{e}} δ_{e}]

(5)

where $\bar{q}$ is the dynamic pressure, S is the wing area, $δ_{e}$ is the elevator deflection, $c_{A}$ is the mean aerodynamic chord and $C_{m *}$ are the pitch moment coefficients. The drag force and lift force are found by

D = \bar{q} S [C_{D 0} + C_{D α} (α - α_{0}) + C_{D δ_{e}} δ_{e}]

(6)

L = \bar{q} S [C_{L 0} + C_{L α} (α - α_{0}) + C_{L δ_{e}} δ_{e}]

(7)

where $C_{D *}$ and $C_{L *}$ are the lift and drag coefficients, respectively. The engine thrust is found by

T = T_{m} δ_{p}

(8)

where $δ_{p}$ is the throttle opening ranging from 0–100% and $T_{m}$ is the maximal thrust.

The following obtains the expressions $F_{c x}$ , $F_{c z}$ and $M_{c}$ . According to the theorem of the composition of particles’ acceleration, the absolute acceleration vector of the cargo is the sum of its implicated acceleration vector, coriolis acceleration vector and relative acceleration vector

a_{c} = \underset{implicated acceleration}{\underset{︸}{\frac{d V}{d t} + \frac{d Ω}{d t} \times r_{c} + Ω \times (Ω \times r_{c})}} + \underset{Coriolis acceleration}{\underset{︸}{2 Ω \times \frac{\tilde{d} r_{c}}{d t^{2}}}} + \underset{relative acceleration}{\underset{︸}{\frac{{\tilde{d}}^{2} r_{c}}{d t^{2}}}}

(9)

where $Ω$ is the angular velocity vector of the aircraft; V is the speed vector of the aircraft; $r_{c}$ is the position vector of point $c$ with respect to point o; and $\tilde{d} (\cdot) / d t$ is the relative derivative operator. Then, $a_{c}$ can be expanded in the wind-axes frame (Liu et al., 2015a) as

a_{w x} = \overset{\cdot}{V} + q^{2} r_{c} \cos α - {\overset{\cdot\cdot}{r}}_{c} \cos α + \overset{\cdot}{q} r_{c} \sin α + 2 q {\overset{\cdot}{r}}_{c} \sin α

(10)

a_{w z} = - V \overset{\cdot}{γ} - q^{2} r_{c} \sin α + {\overset{\cdot\cdot}{r}}_{c} \sin α + \overset{\cdot}{q} r_{c} \cos α + 2 q {\overset{\cdot}{r}}_{c} \cos α

(11)

where $a_{w x}$ and $a_{w z}$ are the x and z components of $a_{c}$ in the wind-axes frame, $r_{c}$ is the distance of point $c$ to point o. As shown in Figure 2, the dynamic equation for the cargo can be obtained as

m_{c} a_{w x} = F_{c x} - F_{p} - m_{c} g \sin γ

(12)

m_{c} a_{w z} = - F_{c z} + m_{c} g \cos γ

(13)

F_{c x} = μ F_{c z}

(14)

where $m_{c}$ is the mass of the cargo, $μ$ is the friction coefficient of rolling between the cargo and the roller on the floor, $F_{p}$ is the extraction force and $F_{p} = m_{c} g λ$ with $λ$ denoting the extraction ratio.

Figure 2.

Forces analysis of the cargo.

Combining equations (10) to (14), we can obtain $F_{c x}$ and $F_{c z}$ as

F_{c x} = F_{p} + m_{c} r_{c} \sin α \dot{q} + m_{c} \dot{V} - (m_{c} g \cos θ - 2 m_{c} q {\dot{r}}_{c}) \sin α + (m_{c} g \sin θ + m_{c} q^{2} r_{c} - m_{c} {\overset{\cdot\cdot}{r}}_{c}) \cos α

(15)

F_{c z} = (m_{c} g \cos θ - 2 m_{c} q {\dot{r}}_{c}) \cos α + m_{c} V \dot{γ} - m_{c} r_{c} \cos α \dot{q} + (m_{c} g \sin θ + m_{c} q^{2} r_{c} - m_{c} {\overset{\cdot\cdot}{r}}_{c}) \sin α

(16)

with

{\overset{\cdot\cdot}{r}}_{c} = \dot{V} \cos α + V \sin α \dot{γ} + g \sin θ - μ g \cos θ + μ F_{p} \sin α / m_{c} + r_{c} q^{2} + F_{p} \cos α / m_{c} + μ (\dot{V} \sin α - V \dot{γ} \cos α + \dot{q} r_{c} + 2 q {\dot{r}}_{c})

(17)

From Figure 1, along with equations (15) and (16), the disturbance moment $M_{c}$ is obtained as

\begin{matrix} M_{c} = r_{c} \cdot (F_{c z} \cos α - F_{c x} \sin α) \\ = m_{c} r_{c} g \cos θ - F_{p} r_{c} \sin α \\ - m_{c} r_{c} (\overset{\cdot}{V} \sin α - V \overset{\cdot}{γ} \cos α + \overset{\cdot}{q} r_{c} + 2 q {\overset{\cdot}{r}}_{c}) \end{matrix}

(18)

Remark 1. It is observed from equations (1) to (3) and (15) to (18) that the aircraft–cargo dynamics are a strong nonlinear system subject to the coupling of the aircraft states with the cargo parameters. Moreover, the system always contains uncertainties, such as aerodynamic data perturbation and uncertain machinery faults. Readers can refer to Liu et al. (2015a) for detailed discussions about the model.

From equations (1) to (8) and (15) to (18), together with the consideration of uncertainties, we can rewrite the aircraft–cargo model to the following compact form

\overset{\cdot}{x} = f (x) + Δ f (x) + B (x) ω u

(19)

where $x = {[V, α, q, θ]}^{T}$ ; $u = {[δ_{e}, δ_{p}]}^{T}$ ; $B (x) = [b_{1}, b_{2}] = {[\begin{matrix} b_{11} b_{21} b_{31} b_{41} \\ b_{12} b_{22} b_{32} b_{42} \end{matrix}]}^{T}$ ; $f (x) = {[f_{1}, f_{2}, f_{3}, f_{4}]}^{T}$ ; $Δ f (x) = [Δ f_{1}, Δ f_{2}, Δ f_{3}, Δ f_{4}]^{T}$ is the uncertainty function; $ω$ is the unknown constant input gain matrix reading as $ω = diag (ω_{11}, ω_{22})$ with $0 < ω_{11}, ω_{22} \leq 1$ . $f_{i}$ and $b_{ij}$ ( $i = 1, 2, 3, 4$ ; $j = 1, 2$ ) are found by

{\begin{matrix} f_{1} = [- \bar{q} S (C_{D 0} + C_{D α} (α - α_{0})) + T_{0} \cos α - m_{b} g \sin γ - m_{c} r_{c} \sin α f_{3} \\ - (Λ_{1} \cos α - Λ_{2} \sin α + F_{p})] / (m_{b} + m_{c}) \\ f_{2} = [- T_{0} \sin α - \bar{q} S (C_{L 0} + C_{L α} (α - α_{0})) + m_{b} g \cos γ + Λ_{1} \sin α + Λ_{2} \cos α + (m_{b} + m_{c}) Vq \\ - m_{c} r_{c} \cos α f_{3}] / [(m_{b} + m_{c}) V] \\ f_{3} = \bar{q} S c_{A} (C_{m 0} + C_{m α} (α - α_{0}) + C_{mq} q c_{A} / 2 V) / Λ_{3} + m_{c} r_{c} [F_{p} \sin α - Λ_{2} - m_{b} g \cos γ \cos α \\ + m_{b} g \sin γ \sin α + \bar{q} S \sin α (C_{D 0} + C_{D α} (α - α_{0})) + \bar{q} S \cos α (C_{L 0} + C_{L α} (α - α_{0}))] / [(m_{b} + m_{c}) Λ_{3}] \\ + r_{c} (Λ_{2} - F_{p} \sin α) / Λ_{3} \\ f_{4} = q \end{matrix}

(20)

{\begin{matrix} b_{11} = - (m_{c} r_{c} \sin α Λ_{4} + \bar{q} S C_{D δ_{e}}) / (m_{b} + m_{c}) \\ b_{12} = T_{m} \cos α / (m_{b} + m_{c}) \\ b_{21} = - (\bar{q} S C_{L δ_{e}} + m_{c} r_{c} \cos α Λ_{4}) / [(m_{b} + m_{c}) V] \\ b_{22} = - T_{m} \sin α / [(m_{b} + m_{c}) V] \\ b_{31} = Λ_{4}, b_{32} = 0, b_{41} = b_{42} = 0 \end{matrix}

(21)

and $Λ_{i}$ ( $i = 1, 2, 3, 4$ ) are as follows

{\begin{matrix} Λ_{1} = m_{c} g \sin θ + m_{c} q^{2} r_{c} - m_{c} {\overset{\cdot\cdot}{r}}_{c} \\ Λ_{2} = m_{c} g \cos θ - 2 m_{c} q {\overset{\cdot}{r}}_{c} \\ Λ_{3} = I_{y} + m_{c} r_{c}^{2} - m_{c}^{2} r_{c}^{2} / (m_{b} + m_{c}) \\ Λ_{4} = \bar{q} S m_{c} r_{c} (C_{D δ_{e}} \sin α + C_{L δ_{e}} \cos α) / [(m_{b} + m_{c}) Λ_{3}] + \bar{q} S c_{A} C_{m δ_{e}} / Λ_{3} \end{matrix}

(22)

The uncertainty functions $Δ f_{i}$ ( $i = 1, 2, 3, 4$ ) introduced by the aerodynamic data perturbation are obtained as

{\begin{matrix} Δ f_{1} = [- \bar{q} S (Δ C_{D 0} + Δ C_{D α} (α - α_{0})) - m_{c} r_{c} \sin α Δ f_{3}] / (m_{b} + m_{c}) \\ Δ f_{2} = [- \bar{q} S (Δ C_{L 0} + Δ C_{L α} (α - α_{0})) - m_{c} r_{c} \cos α Δ f_{3}] / [(m_{b} + m_{c}) V] \\ Δ f_{3} = \bar{q} S c_{A} (Δ C_{m 0} + Δ C_{m α} (α - α_{0}) + Δ C_{mq} q c_{A} / 2 V) / Λ_{3} \\ + m_{c} r_{c} \bar{q} S [\sin α (Δ C_{D 0} + Δ C_{D α} (α - α_{0})) + \cos α (Δ C_{L 0} + Δ C_{L α} (α - α_{0}))] / [(m_{b} + m_{c}) Λ_{3}] \\ Δ f_{4} = 0 \end{matrix}

(23)

Here, $Δ C_{L *}$ , $Δ C_{D *}$ and $Δ C_{m *}$ are the perturbation of the lift, drag and pitch moment coefficients, respectively. We introduce the following notation

{\begin{matrix} E_{11} = - \bar{q} {Sm}_{c}^{2} r_{c}^{2} \sin α \cos α / [{(m_{b} + m_{c})}^{2} Λ_{3}] \\ E_{12} = - \bar{q} S / (m_{b} + m_{c}) - \bar{q} {Sm}_{c}^{2} r_{c}^{2} \sin^{2} α / [{(m_{b} + m_{c})}^{2} Λ_{3}] \\ E_{13} = - \bar{q} S c_{A} m_{c} r_{c} \sin α / [(m_{b} + m_{c}) Λ_{3}] \\ E_{21} = - \bar{q} S / [(m_{b} + m_{c}) V] - \bar{q} {Sm}_{c}^{2} r_{c}^{2} \cos^{2} α / [{(m_{b} + m_{c})}^{2} Λ_{3} V] \\ E_{22} = - \bar{q} {Sm}_{c}^{2} r_{c}^{2} \sin α \cos α / [{(m_{b} + m_{c})}^{2} Λ_{3} V] \\ E_{23} = - \bar{q} S c_{A} m_{c} r_{c} \cos α / [(m_{b} + m_{c}) Λ_{3} V] \\ E_{31} = \bar{q} S m_{c} r_{c} \cos α / [(m_{b} + m_{c}) Λ_{3}] \\ E_{32} = \bar{q} S m_{c} r_{c} \sin α / [(m_{b} + m_{c}) Λ_{3}] \\ E_{33} = \bar{q} S c_{A} / Λ_{3} \end{matrix}

(24)

φ = [\begin{matrix} 1 & α - α_{0} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & α - α_{0} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & α - α_{0} & q c_{A} / 2 V \end{matrix}]

(25)

P = [Δ C_{L 0}, Δ C_{L α}, Δ C_{D 0}, Δ C_{D α}, Δ C_{m 0}, Δ C_{m α}, Δ C_{mq}]^{T}

(26)

Then, $Δ f_{1}$ , $Δ f_{2}$ and $Δ f_{3}$ can be written as

Δ f_{1} = [E_{11}, E_{12}, E_{13}] φ P

(27)

Δ f_{2} = [E_{21}, E_{22}, E_{23}] φ P

(28)

Δ f_{3} = [E_{31}, E_{32}, E_{33}] φ P

(29)

Input–output feedback linearization

Let $y = [y_{1}, y_{2}]^{T} = [V, θ]^{T}$ be the output of system (19). Using the input–output feedback linearization method, we can transform system (19) to the following form

[y_{1}^{(r_{1})}, y_{2}^{(r_{2})}]^{T} = A (x) + Δ A (x) + D (x) u

(30)

where $r_{i}$ is the relative degree of $y_{i}$ , which satisfies $L_{b_{k}} (L_{f}^{l - 1} y_{i}) = 0, (l = 1, 2, \dots, r_{i} - 1)$ and $L_{b_{k}} (L_{f}^{r_{i} - 1} y_{i}) \neq 0$ with $L (\cdot)$ being the Lie derivative operator (Slotine and Li, 1991). Calculation shows that $r_{1} = 1$ , $r_{2} = 2$ . The decoupling matrix is found by

D (x) = [\begin{matrix} L_{ω_{11} b_{1}} L_{f}^{r_{1} - 1} y_{1} L_{ω_{22} b_{2}} L_{f}^{r_{1} - 1} y_{1} \\ L_{ω_{11} b_{1}} L_{f}^{r_{2} - 1} y_{2} L_{ω_{22} b_{2}} L_{f}^{r_{2} - 1} y_{2} \end{matrix}] = [\begin{matrix} ω_{11} b_{11} ω_{22} b_{12} \\ ω_{11} b_{31} ω_{22} b_{32} \end{matrix}]

(31)

Let $B_{ω} (x) = [\begin{matrix} b_{11} b_{12} \\ b_{31} b_{32} \end{matrix}]$ , then $D (x)$ can be written as

D (x) = B_{ω} (x) ω

(32)

From $- π / 2 < α < π / 2$ , we have $| b_{11} b_{32} | << | b_{12} b_{31} |$ (Li et al., 2013; Liu et al., 2015a), which, together with $0 < ω_{11}, ω_{22} \leq 1$ , implies that $D (x)$ is nonsingular. $A (x)$ is given by

A (x) = [L_{f}^{r_{1}} y_{1}, L_{f}^{r_{2}} y_{2}]^{T} = [f_{1}, f_{3}]^{T}

(33)

and matrix $Δ A (x)$ is obtained as

Δ A (x) = [L_{Δ f} L_{f}^{r_{1} - 1} y_{1}, L_{Δ f} L_{f}^{r_{2} - 1} y_{2}]^{T} = [Δ f_{1}, Δ f_{3}]^{T} = E_{P} (x) P

(34)

with $E_{P} (x) = [\begin{matrix} E_{11} E_{12} E_{13} \\ E_{31} E_{32} E_{33} \end{matrix}] φ$ .

From equation (31), (32) and (34) we can rewrite system (30) as

[y_{1}^{(r_{1})}, y_{2}^{(r_{2})}]^{T} = A (x) + E_{P} (x) P + B_{ω} (x) ω u

(35)

Remark 2. The sliding-mode controller requires the upper bounds of the uncertainties ( $Δ f (x)$ or $Δ A (x)$ ) to specify the control gains to satisfy the requirement of stability and robustness (Li et al., 2013; Liu et al., 2015a; Yang and Lu, 2012; Zhang and Shi, 2009b). However, the nonlinear uncertainties, composed of aircraft–cargo dynamics coupled with aerodynamic coefficients’ perturbation, are difficult to achieve in advance. In this case, the feedback gains need to be set large enough, which in turn can worsen the chattering problem. In this study, the nonlinearity $Δ A (x)$ is factorized into a known matrix (i.e. $E_{P} (x)$ ) and an uncertainty function (i.e. P ), and we use adaptation methods to approximate the function.

Control law design

The airdrop process starts as the aircraft glides and adjusts to a steady wings-level flight state. At the dropping point, the cargo is pulled out of the tank by the extraction umbrella. To guarantee airdrop precision and flight safety, the autopilot should stabilize the flight state at the trim position as far as possible (Feng et al., 2011; Li et al., 2013; Liu et al., 2015a, 2015b; Yang and Lu, 2012; Zhang and Shi, 2009b).

Firstly, the inner-loop adaptive sliding-mode controller is designed to stabilize the aircraft’s speed and pitch angle. Let $V_{d}$ and $θ_{d}$ denote the trim speed and pitch angle, and define the tracking errors as

e = [e_{1}, e_{2}]^{T} = [V - V_{d}, θ - θ_{d}]^{T}

(36)

Define the sliding surface as

s_{1} = e_{1}

(37)

s_{2} = {\overset{\cdot}{e}}_{2} + c e_{2}

(38)

where $c > 0$ is a tuning parameter. Letting $S = [s_{1}, s_{2}]^{T}$ , taking the time derivative of S yields

\overset{\cdot}{S} = A (x) + E_{P} (x) P + B_{ω} (x) ω u - Y_{d} + C_{e}

(39)

where $Y_{d} = [{\overset{\cdot}{V}}_{d}, {\overset{\cdot\cdot}{θ}}_{d}]^{T}$ and $C_{e} = [0, c (\overset{\cdot}{θ} - {\overset{\cdot}{θ}}_{d})]^{T}$ .

The control law is designed as

u = - [B_{ω} (x) \hat{ω}]^{- 1} [A (x) + E_{P} (x) \hat{P} - Y_{d} + C_{e} + ε sgn (S) + KS]

(40)

where $ε, K > 0$ are tuning parameters represented as $ε = diag (ε_{1}, ε_{2})$ and $K = diag (k_{1}, k_{2})$ ; $\hat{ω}$ and $\hat{P}$ are the estimation of $ω$ and P , respectively, which are governed by

\overset{\cdot}{\hat{P}} (t) = Γ_{p} Proj (\hat{P}, E_{p}^{T} S)

(41)

\overset{\cdot}{\hat{ω}} (t) = Γ_{ω} Proj (\hat{ω}, B_{ω}^{T} S u^{T})

(42)

where $Γ_{p}$ and $Γ_{ω}$ are symmetric positive definite matrixes and $Proj (\cdot, \cdot)$ denotes the projection operator (Appendix 1), which ensures that $\hat{ω} (t) \in Ω$ .

Remark 3. The projection-based adaptation law ensures that $\hat{ω} (t)$ is nonsingular. For instance, we can set

Ω = [\begin{matrix} [0.5, 1] & [0, 0.01] \\ [0, 0.01] & [0.5, 1] \end{matrix}]

(43)

Since $\hat{ω} (t) \in Ω$ , thus $\hat{ω} (t)$ is always nonsingular. This further implies that $B_{ω} (x) \hat{ω}$ is nonsingular.

Theorem 1. Given the system in equation (35) controlled by equation (40) with the adaptation laws defined via equations (41) and (42), the estimation errors of the uncertainties are uniformly bounded and the tracking errors converge to zeroes asymptotically, i.e.

lim_{t \to \infty} e_{i} (t) = 0 (i = 1, 2)

(44)

Proof: Consider the positive definite function

σ (S, \tilde{P}, \tilde{ω}) = \frac{1}{2} S^{T} S + \frac{1}{2} {\tilde{P}}^{T} Γ_{p}^{- 1} \tilde{P} + \frac{1}{2} trace ({\tilde{ω}}^{T} Γ_{ω}^{- 1} \tilde{ω})

(45)

where $\tilde{P} = \hat{P} - P$ and $\tilde{ω} = \hat{ω} - ω$ are the estimation errors of the uncertainties. Taking the time derivative of $σ$ yields

\begin{matrix} \overset{\cdot}{σ} = S^{T} \overset{\cdot}{S} + {\tilde{P}}^{T} Γ_{p}^{- 1} \overset{\cdot}{\hat{P}} + trace ({\tilde{ω}}^{T} Γ_{ω}^{- 1} \overset{\cdot}{\hat{ω}}) \\ = S^{T} [A (x) + E_{P} (x) P + B_{ω} (x) ω u - Y_{d} + C_{e}] \\ + {\tilde{P}}^{T} Γ_{p}^{- 1} \overset{\cdot}{\hat{P}} + trace ({\tilde{ω}}^{T} Γ_{ω}^{- 1} \overset{\cdot}{\hat{ω}}) \\ = - S^{T} ε sgn (S) - S^{T} KS - S^{T} E_{P} (x) \tilde{P} - S^{T} B_{ω} \tilde{ω} u \\ + {\tilde{P}}^{T} Γ_{p}^{- 1} \overset{\cdot}{\hat{P}} + trace ({\tilde{ω}}^{T} Γ_{ω}^{- 1} \overset{\cdot}{\hat{ω}}) \end{matrix}

(46)

From $S^{T} B_{ω} \tilde{ω} u = trace (u S^{T} B_{ω} \tilde{ω}) = trace ({\tilde{ω}}^{T} B_{ω}^{T} S u^{T})$ , it follows that

\dot{σ} = - S^{T} ε sgn(S) - S^{T} KS + {\tilde{P}}^{T} (Γ_{p}^{- 1} \dot{\hat{P}} - E_{p}^{T} S) + trace [{\tilde{ω}}^{T} (Γ_{ω}^{- 1} \dot{\hat{ω}} - B_{ω}^{T} S u^{T})]

(47)

Substituting equations (41) and (42) into equation (47) yields

\dot{σ} = - S^{T} ε sgn(S) - S^{T} KS + {\tilde{P}}^{T} [Γ_{p}^{- 1} Γ_{p} Proj(\hat{P}, E_{p}^{T} S) - E_{p}^{T} S] + trace [{\tilde{ω}}^{T} (Γ_{ω}^{- 1} Γ_{ω} Proj(\hat{ω}, B_{ω}^{T} S u^{T}) - B_{ω}^{T} S u^{T})]

(48)

Using Property A2 of the projection operator, we can obtain that

{\tilde{P}}^{T} [Proj (\hat{P}, E_{p}^{T} S) - E_{p}^{T} S] \leq 0

(49)

{\tilde{ω}}^{T} [Proj (\hat{ω}, B_{ω}^{T} S u^{T}) - B_{ω}^{T} S u^{T}] \leq 0

(50)

which further leads to

\overset{\cdot}{σ} \leq - S^{T} ε sgn (S) - S^{T} KS \leq 0

(51)

Therefore, we can conclude that S , $\tilde{P}$ , $\tilde{ω}$ are uniformly bounded. From equations (45) and (51) we can also conclude that because $σ (t)$ is bounded from below and is non-increasing with time, it has a limit, i.e. $lim_{t \to \infty} σ (t) = σ_{\infty}$ . It then further follows from equation (51) that

\int_{0}^{\infty} S^{T} KS d τ \leq σ_{0} - σ_{\infty}

(52)

where ${σ (t) |}_{t = 0} = σ_{0}$ . Because $λ_{min} (K) ‖ S ‖^{2} \leq S^{T} KS$ , it follows that $\int_{0}^{\infty} {‖ S ‖}^{2} d τ$ is bounded. While $\overset{\cdot}{S}$ is also bounded, we can conclude from Barbalat’s lemma that $s_{1} (t), s_{2} (t) \to 0$ as $t \to \infty$ . Hence, $e_{1} (t), e_{2} (t) \to 0$ as $t \to \infty$ . The proof is thus completed.

In the end, the outer loop uses the classical PID control method to hold the aircraft’s altitude. Figure 3 illustrates the framework of the control system, where H is the flight altitude; $H_{d}$ is the desired command of H; and $K_{p}$ , $K_{I}$ and $K_{D}$ are the proportion, integration and differentiation gains, respectively.

Figure 3.

Framework of control system.

Simulation analysis

The following simulations were conducted on a 24,955 kg transport aircraft performing an 8000 kg load airdrop task. The cargo is initially locked in the centre of gravity (C.G.) of the aircraft. The aircraft is trimmed at the following condition: $H_{0} = 25 m$ , $V_{0} = 70 m / s$ and $α_{0} = θ_{0} = 5.9813 \circ$ with $δ_{p} = 34.1 %$ , $δ_{e} = 0 \circ$ , the horizontal stabilizer $δ_{h} = - 5.4093 \circ$ and the flap $δ_{f} = 25 \circ$ . The extraction ratio is set as $λ = 0.2$ .

With the requirements of mission completeness and flight safety, the performance indexes for the airdrop operations are given as (Li et al., 2013): (1) $| Δ H | \leq 13 m$ and $H > 6 m$ ; (2) $| Δ V | \leq 0.13 V_{0}$ ; (3) $| Δ θ | \leq 5 \circ$ and $θ > 2 \circ$ ; and (4) $α \leq 0.7 α_{s}$ with $α_{s}$ denoting the stalling angle of attack.

To test the performance and robustness of the control system, the following three cases are simulated and compared:

Case 1: Hypothesize the dynamic model is accurate, i.e. $P = [0000000]^{T}$ and $ω = diag (1, 1)$ .

Case 2: The aerodynamic coefficients perturb −20% and the actuator fault is 20%, i.e. $P = - 20 % \cdot {[C_{L 0}, C_{L α}, C_{D 0}, C_{D α}, C_{m 0}, C_{m α}, C_{m q}]}^{T}$ and $ω = diag (0.8, 0.8)$ .

Case 3: The aerodynamic coefficients perturb 20% and the actuator fault is 20%, i.e. $P = 20 % \cdot {[C_{L 0}, C_{L α}, C_{D 0}, C_{D α}, C_{m 0}, C_{m α}, C_{m q}]}^{T}$ and $ω = diag (0.8, 0.8)$ .

The compact set $Ω$ can be conservatively set as the one in equation (43). The inner-loop controller parameters are set as $c = 2$ , $ε_{1} = 0.6$ , $ε_{2} = 0.1$ , $k_{1} = 0.2$ , $k_{2} = 1$ , $Γ_{ω} = diag (3, 4)$ and $Γ_{p} = diag (5.5, 2, 2, 1, 7, 10, 15)$ after experimental tuning. The outer-loop altitude-hold controller parameters are set as $K_{p} = 0.2$ , $K_{I} = 0.23$ and $K_{D} = 0.0027$ . Successful implementation of the control method can be observed from Figure 4. In all three cases, the altitude increment is controlled in the range of 0.2 m and, after the cargo is dropped, the altitude is well maintained at the predefined trim position within 6 s. Owing to the loss of the heavy weight, the final angle of attack and the final pitch angle become smaller compared with that of the trim position. But the change value is less than 5°, and the final pitch angle is greater than 2°, which could meet the performance indexes. Figure 5 shows that the elevator deflection and the throttle opening are within the magnitude limits, and there is no severe chattering problem. In summary, the pitch-up motion of the aircraft caused by the moving cargo is suppressed effectively through appropriately configuring the elevator and the throttle.

Figure 4.

Aircraft responses of the dropping process with the proposed control method (Cases 1, 2 and 3).

Figure 5.

Curves of the throttle opening and elevator deflection (Cases 1, 2 and 3).

We further make a comparison of control performance between the proposed control method and the iterative quasi-sliding-mode control (IQSMC) method (Liu et al., 2015a), in the presence of −20% aerodynamic coefficients’ uncertainty and 20% actuator fault. The iterative quasi-sliding-mode controller is recalled from Liu et al. (2015a) as the following

\begin{array}{l} u = {[\begin{matrix} b_{11} & b_{12} \\ b_{31} & b_{32} \end{matrix}]}^{- 1} (- [\begin{matrix} f_{1} \\ f_{3} \end{matrix}] + [\begin{matrix} {\dot{V}}_{d} \\ {\overset{\cdot\cdot}{θ}}_{d} \end{matrix}] - [\begin{matrix} 0 \\ ζ_{2} (\dot{θ} - {\dot{θ}}_{d}) \end{matrix}] + [\begin{matrix} - ξ_{1} e^{- ξ_{1} t} (V (0) - V_{d} (0)) \\ - ξ_{2} e^{- ξ_{2} t} [ζ_{2} (θ (0) - θ_{d} (0)) + \dot{θ} (0) - {\dot{θ}}_{d} (0)] \end{matrix}] \\ - [\begin{matrix} k_{1} s_{11} e^{- λ_{1} | s_{11} |} \\ k_{2} s_{12} e^{- λ_{2} | s_{12} |} \end{matrix}] - [\begin{matrix} η_{1} sat (‖ S_{2} ‖ / β) \cdot (s_{21} / ‖ S_{2} ‖) \\ η_{2} sat (‖ S_{2} ‖ / β) \cdot (s_{22} / ‖ S_{2} ‖) \end{matrix}]) \end{array}

(53)

where $ξ_{1}, ξ_{2}, λ_{1}, λ_{2} > 0$ are exponentially approaching constants; $η_{1}$ , $η_{2}$ are robust terms that need to be chosen larger than the bounds of the system uncertainties; $k_{1}$ , $k_{2}$ and $ζ_{2}$ are positive constants; $s_{11}$ , $s_{12}$ , $s_{21}$ and $s_{22}$ are sliding manifolds with $S_{2} = [s_{21}, s_{22}]^{T}$ ; $β$ is the boundary layer thickness; and $sat (\cdot)$ is defined as

sat (\frac{‖ S_{2} ‖}{β}) \frac{S_{2}}{‖ S_{2} ‖} = {\begin{matrix} sgn (S_{2}) & If ‖ S_{2} ‖ \geq β \\ \frac{S_{2}}{β} & If ‖ S_{2} ‖ < β \end{matrix}

with $sgn (\cdot)$ being the sign function. We set $ξ_{1} = ξ_{2} = 5$ , $ζ_{2} = 2$ , $k_{1} = 3$ , $k_{2} = 2$ , $λ_{1} = 1$ , $λ_{2} = 3.3$ , $η_{1} = 3$ , $η_{2} = 0.1$ and $β = 2$ after referring to Liu et al. (2015a). The outer-loop altitude-hold controller parameters are set as those of the proposed method. Figure 6 illustrates that the altitude and the speed are well maintained at about 6 s for both of the controllers, in the presence of uncertainties, and the results meet the airdrop performance indexes. However, the curves of the throttle opening and elevator deflection imply that the IQSMC method might lead to severe chattering. This is unfavourable for real applications. When using the proposed controller, the undesired chattering is reduced significantly. This is because the projection-based adaptation technique is employed to estimate the uncertainties, and the uncertainties can be compensated for by the estimated value in the control input. To summarize, the proposed method not only guarantees tracking performance but also yields a moderate control behaviour.

Figure 6.

Responses comparison of the dropping process with the proposed method and the IQSMC method (in the presence of −20% aerodynamic coefficients’ uncertainty and 20% actuator fault).

Conclusions

This paper proposes a novel flight control method that combines output decoupling SMC with projection-based adaptive function approximation for heavyweight airdrop operations. The controller is capable of tracking the airspeed and pitch angle commands, while being robust to unknown but bounded system uncertainties caused by aerodynamic perturbation as well as actuator faults that might occur during flight. The method uses projection-based adaptation strategies to achieve robustness against uncertainties, and this overcomes the conservation of the SMC method that relies on the knowledge of the bounds on the complex uncertainties. The stability properties are proved using Lyapunov’s theories. Simulations verify that the performance of the controller can meet the airdrop mission performance indexes well, even in the presence of ±20% aerodynamic coefficients’ uncertainty and 20% actuator fault. The research will benefit future practical airdrop missions.

The presentation of this paper has assumed that all of the flight states are measurable and the uncertainties are slowly time-varying or time-invariant. This is, of course, an idealized situation. The investigation developing this method to output feedback systems is interesting. The sliding-mode parameters and the adaptation gains are tuned in a trial-and-error way, which is tedious work. A criterion to specify the control parameters is needed, and this will be one of our future studies.

Footnotes

Appendix 1

The projection operator introduced in Pomet et al. (1992) bounds the estimated parameters by definition. We recall the main definitions from Pomet et al. (1992):

Definition 1. Consider a convex compact set with a smooth boundary given by

(54)

Ω_{c} = {θ \in R^{n} | f (θ) \leq c}, 0 \leq c \leq 1

where $f : R^{n} \to R$ is the following smooth convex function

(55)

f (θ) = \frac{θ^{T} θ - θ_{max}^{2}}{ε_{θ}}

where $θ_{max}$ is the norm bound imposed on the vector $θ$ , and $0 < ε_{θ} < 1$ stands for the projection tolerance bound of our choice. For any given $y \in R^{n}$ , the projection operator is defined as

(56)

Proj (θ, y) = {\begin{matrix} y, if f (θ) < 0 \\ y, if f (θ) \geq 0, \nabla f^{T} y \leq 0 \\ y - g (f, y), & if f (θ) \geq 0, \nabla f^{T} y > 0 \end{matrix}

where $\nabla f (θ)$ is the gradient vector of $f (\cdot)$ evaluated at $θ$ , and

(57)

g (f, y) = \frac{\nabla f \nabla f^{T} y}{{‖ \nabla f ‖}^{2}} f (θ)

The properties of the projection operator are given by the following lemma

Lemma 1. Let

(58)

θ^{*} \in Ω_{0} = {θ \in R^{n} | f (θ) \leq 0}

and let the parameter $θ (t)$ evolve according to the following dynamics

(59)

\overset{\cdot}{θ} (t) = Proj (θ (t), y), θ (t_{0}) \in Ω_{c}

Then

(60)

θ (t) \in Ω_{c}

for all $t \geq t_{0}$ , and

(61)

{(θ - θ^{*})}^{T} (Proj (θ, y) - y) \leq 0

Property 1. The projection operator $Proj (θ, y)$ does not alter y if $θ$ belongs to the set $Ω_{0}$ . In the set ${0 \leq f (θ) \leq 1}$ , $Proj (θ, y)$ subtracts a vector normal to the boundary of $Ω_{c}$ so that we get a smooth transformation from the original vector field y to an inward or tangent vector field for $c = 1$ . Therefore, on the boundary of $Ω_{c}$ , $\overset{\cdot}{θ} (t)$ always points toward the inside of $Ω_{c}$ and $θ (t)$ never leaves the set $Ω_{c}$ .

Property 2. From the convexity of function $f (θ)$ , it follows that for any $θ^{*} \in Ω_{0}$ and $θ \in Ω_{c}$ , the inequality

(62)

{(θ - θ^{*})}^{T} \nabla f (θ) \leq 0

holds. It then follows from Definition 1 that

(63)

{(θ - θ^{*})}^{T} (Proj (θ, y) - y) = {\begin{cases} 0, & if f (θ) < 0 \\ 0, & if f (θ) \geq 0, \nabla f^{T} y \leq 0 \\ \frac{\underset{\leq 0}{\underset{︸}{{(θ - θ^{*})}^{T} \nabla f (θ)}} \underset{\geq 0}{\underset{︸}{\nabla f^{T} y}} \underset{\geq 0}{\underset{︸}{f (θ)}}}{{‖ \nabla f ‖}^{2}}, & if f (θ) \geq 0, \nabla f^{T} y > 0 \end{cases}

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (Grant No. 61273141) and the Aviation Science Foundation of China (Grant No. 20141396012).

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