Backstepping control based on L 1 adaptive theory for large transport aircraft heavy load airdrop

Abstract

A study of the L₁ adaptive controller is conducted based on the backstepping method for the model of large transport aircraft with drastic changes appearing in heavy load airdrop process. The system is divided into an attitude subsystem and a velocity subsystem. For the attitude subsystem, the backstepping control is used to design the virtual control of path angle and the pitch angle in external loop with the L₁ adaptive controller designed in internal loop to estimate the uncertainties and disturbances in the subsystem and to compensate them. In the stability analysis, the uniform boundedness of all signals in the closed-loop system is proven. Simulation results show that the proposed control method preserves the quick dynamic torque response, high efficiency, and robustness in heavy load airdrop; to some extent it can alleviate the control switch lead or lag problem and ensure the safety of the transport aircraft to fulfill the complete airdrop mission.

Keywords

Heavy load airdrop backstepping uncertainties virtual control

Introduction

Large air transport and precision airdrop plays an important role in military and rescue operation, with the airdrop ability being an important indicator of national defense force and the significant means for rapid response and long distance maneuvers in a modern war.

However, the robust and precise airdrop is difficult to carry out due to model mutation occurring especially during the large transport heavy drop. The aircraft controller cannot be switched accurately during airdrop, that is to say, the switch can occur either ahead or lagged, causing intense oscillation or machine apart. This problem belongs to the mutation model control,^{1
–3} and it is necessary to solve it properly for the safety of large transport aircrafts.

Currently, the study of transport load airdrop focuses on the analysis,^4,5 modeling,⁶ simulation,^7,8 and control law design.^6,9 In the study by Jann,⁸ the general military airdrop process of longitudinal channel is simulated, with the influence of airdrop on the flight control system studied by Chen et al.^6,9 At the moment of cargo separating from aircraft, the large angle of attack motion and the problem from controlling models of security are shown in the literatures.^{10

–13} The variable structure control in Xu and Sun¹⁴ and sliding mode control in Zhang and Shi¹⁵ are also studied to ensure the safety of the aircraft.

With the development of the modern control theory, many methods^{16

–21} are useful in improving the robustness of the system. In earlier works,^17,20 the discrete adaptive backstepping is studied for a class of uncertain systems and they are also applied in helicopter control by Li et al.²² To deal with the calculation explosion, the dynamic surface control and instruction filter are introduced in the study by Xu.²³ The backstepping combined with sliding control is introduced in the longitudinal sliding mode adaptive controller for the unmatched and uncertain longitudinal parameters. The L₁ adaptive control theory is presented by Cao and Hovakimyan^24,25 and widely used later. Compared to the conventional adaptive control, L₁ adaptive control can make the regulation time shorter and ensure better asymptotic tracking and transient performance in the delay system.^{26

–29}

For the instability of control system by model mutation during the ultra low altitude airdrop, this article presents a method combined with backstepping and the L₁ adaptive method for obtaining the robust and precision airdrop. The backstepping method is used for the controller of track angle and pitching angle in the attitude subsystem. The L₁ adaptive controller is carried on the estimation and compensation to improve the dynamic performance and robustness. Through the design of virtual control, compensation is made to solve the problem of calculation explosion, ensuring the stability and reliability. Simulation results show the effectiveness and strong robustness of the method proposed in this article.

The article structure is as follows. The airdrop model is presented in the dynamic model section, the design of controller is given in the third section, an analysis of stability is made in the fourth section, in fifth section, simulation results show the effectiveness of the control method, and the conclusion is drawn in the last section.

Dynamic model

In this article, the force in the airdrop process of the aircraft is analyzed and for the cargo movement displacement as additional state variables, the dynamic model from Christopher et al.²⁶ can be described as follows

\begin{array}{l} \dot{V} = \frac{(T + F_{b} - m_{b} (\ddot{d} - d q^{2})) cos α + m_{b} (2 \dot{d} q + d \dot{q}) sin α}{m} \\ - \frac{W}{m} - g sin γ \\ \dot{α} = \frac{m_{b} (2 \dot{d} q + d \dot{q}) cos α - A - (T + F_{b} - m_{b} (\ddot{d} - d q^{2})) sin α}{m V} \\ + \frac{g cos γ}{V} + q \\ \dot{θ} = q \\ \dot{q} = \frac{M}{I_{y}} \\ \dot{H} = V sin γ \\ \ddot{d} = d q^{2} + μ (2 \dot{d} q + d \dot{q}) + \frac{F_{b}}{m_{b}} \\ + \frac{(A (- sin α + μ cos α) + W (cos α + μ sin α) c_{α} + T)}{m_{a}} \end{array}

In equation (1), V is the plane velocity; α is the attack angle; θ is the pitch angle; H is the altitude; q is the pitch rate; d is the distance of the first cargo moved away; γ is the flight path angle and γ = θ − α; F_b is the traction for cargos movement and $F_{b} = - k_{u} m_{b} g$ ; m is the sum of the mass of aircraft and cargos and $m = m_{a} + m_{b}$ , where m_a is the aircraft mass and m_b is the cargo mass; T is the push force and $T = T_{max} δ_{T}$ , where T_max is the maximum of the push force and δ_T is fuel equivalence ratio; A is the lift force and $A = C_{A} Q S$ ; W is the drag force and $W = C_{W} Q S$ ; Q is the kinetic pressure and $Q = \frac{1}{2} ρ V^{2}$ ;S is the wing area; C_A is the coefficient to lift; C_W is the coefficient to drag, and

C_{A} = C_{A 0} + C_{A α} (α - α_{0}) + C_{A \dot{α}} \frac{\dot{α} c_{A}}{2 V} + C_{A q} \frac{q c_{A}}{2 V} + C_{A V} (V - V_{0}) + C_{A δ_{e}} δ_{e} C_{W} = C_{W 0} + C_{W α} (α - α_{0}) + C_{W V} (V - V_{0}) + C_{W δ_{e}} δ_{e}

In the dynamic model, $M = M_{A} + M_{B} + M_{T}$ , M_A is the pitch moment, M_B is the additional moment caused by the moving cargo, and M_T is the thrust moment. They are expressed as

\begin{array}{l} M_{A} = Q S c_{A} [C_{M 0} + C_{M α} (α - α_{0}) + C_{m V} (V - V_{0}) \\ + C_{M \dot{α}} \frac{\dot{α} c_{A}}{2 V} + C_{M q} \frac{q c_{A}}{2 V} + C_{M δ_{e}} δ_{e}] \\ M_{B} = - n d (A cos α + W sin α) \\ M_{T} = z_{p} T \end{array}

where n is the ratio of m_b and m, I_y is the total inertia of plane and the rest load. The values and units are listed in Tables 1 to 5.

Table 1.

Miscellaneous coefficient values.

Coefficient	Value	Units
S	$5.500 \times 10^{3}$	${ft}^{2} \cdot {ft}^{- 1}$
ρ	$1.225 \times 10^{0}$	$slugs \cdot {ft}^{- 3}$
$I_{y y}$	$3.050 \times 10^{7}$	$lb \cdot ft$
m_a	$1.753 \times 10^{4}$	$lb \cdot {ft}^{- 1}$
g	$3.200 \times 10^{1}$	$ft \cdot s^{- 2}$
$I_{y y b}$	$2.3627 \times 10^{5}$	$lb \cdot ft$
m_b	$2.6038 \times 10^{3}$	$lb \cdot {ft}^{- 1}$
α ₀	$0.0349 \times 10^{0}$	$rad$
μ	$0.01 \times 10^{0}$	—
k_u	$0.1655 \times 10^{0}$	—

Table 2.

Lift coefficient values.

Coefficient	Value	Units
$C_{A 0}$	$2.030 \times 10^{0}$	−
$C_{A α}$	$8.000 \times 10^{0}$	${rad}^{- 1}$
$C_{A \dot{α}}$	$6.700 \times 10^{0}$	${rad}^{- 1} \cdot s$
$C_{A V}$	$0.220 \times 10^{0}$	${ft}^{- 1} \cdot s$
$C_{A q}$	$5.650 \times 10^{0}$	${rad}^{- 1} \cdot s$
$C_{A δ_{e}}$	$1.110 \times 10^{0}$	${rad}^{- 1}$

Table 3.

Drag coefficient values.

Coefficient	Value	Units
$C_{W 0}$	$0.263 \times 10^{0}$	−
$C_{W α}$	$1.130 \times 10^{0}$	${rad}^{- 1}$
$C_{W V}$	0	${ft}^{- 1} \cdot s$
$C_{W δ_{e}}$	0	${rad}^{- 1}$

Table 4.

Moment coefficient values.

Coefficient	Value	Units
z_T	0	ft
c_A	$2.730 \times 10^{1}$	ft
$C_{M 0}$	$1.8979 \times 10^{- 1}$	−
$C_{M α}$	$- 1.450 \times 10^{0}$	${rad}^{- 1}$
$C_{M \dot{α}}$	$- 3.300 \times 10^{0}$	${rad}^{- 1} \cdot s$
$C_{M V}$	$0.071 \times 10^{0}$	${ft}^{- 1} \cdot s$
$C_{M q}$	$- 2.140 \times 10^{1}$	${rad}^{- 1} \cdot s$
$C_{M δ_{e}}$	$- 4.400 \times 10^{0}$	${rad}^{- 1}$

Table 5.

Thrust coefficient values.

Coefficient	Value	Units
T _max	$2.190 \times 10^{5}$	$lb \cdot s^{- 2}$

Remark 1

When all the cargos are dropped, the system model from Christopher et al.²⁶ is

\begin{array}{l} \dot{V} = \frac{T cos α - W}{m} - g sin γ \\ \dot{α} = - \frac{T sin α + A}{m V} + \frac{g cos γ}{V} + q \\ \dot{θ} = q \\ \dot{q} = \frac{M}{I_{y}} \\ \dot{H} = V sin γ \end{array}

Controller design

As the track angle γ is little, it is assumed that $sin γ = γ$ . In the $H - γ$ subsystem, the track error is defined as $\tilde{H} = H - H_{r}$ , and then the instruction for the track angle can be calculated as

γ_{d} = arc sin [\frac{- k_{P} (H - H_{r}) - k_{I} \int (H - H_{r}) d t + k_{D} {\dot{H}}_{r}}{V}]

where the PID parameters k_P, k_I, and k_D can be set by the designer.

When the track angle approaches the expected γ_d, a height index tracking error converges to zero. Through the instruction transformation, the height control can be achieved through the attitude control.

Define the state variables $x = {[\begin{matrix} γ & θ & q \end{matrix}]}^{T}$ . The dynamic equations of attitude subsystem can be written as

{\begin{cases} {\dot{x}}_{1} = f_{1} + g_{1} x_{2} \\ {\dot{x}}_{2} = f_{2} + g_{2} x_{3} \\ {\dot{x}}_{3} = f_{3} + g_{3} u \\ y = x \\ u = δ_{e} \end{cases}

Velocity subsystem can be written in the following form

{\begin{cases} \dot{V} = f_{V} + g_{V} u_{V} \\ y = V \\ u_{V} = δ_{T} \end{cases}

where $\begin{array}{l} f_{1} = - \frac{m_{b} (2 \dot{d} q + d \dot{q}) cos α - (A - C_{A α} θ Q S)}{m V} - \\ \frac{(T + F_{b} - F_{b} - m_{b} (\ddot{d} - d q^{2})) sin α}{m V} - \frac{g cos γ}{V} \end{array}$ , $g_{1} = \frac{C_{A α} Q S}{m V}$ , $f_{2} = 0$ , $g_{2} = 1$ , $f_{3} = \frac{M}{I_{y}} - \frac{C_{M δ_{e}} δ_{e} Q S c_{A}}{I_{y}}$ , $g_{3} = \frac{C_{M δ_{e}} Q S c_{A}}{I_{y}}, \begin{matrix} \begin{array}{l} f_{V} = \frac{(F_{b} - m_{b} (\ddot{d} - d q^{2})) cos α + m_{b} (2 \dot{d} q + d \dot{q}) sin α}{m} -, \\ \frac{W}{m} - g sin γ, \end{array} & g_{V} = \frac{T_{max} cos α}{m} \end{matrix}$

Attitude subsystem model

The attitude subsystem was composed of track angle parts, pitching angle parts, and pitching angle rate parts. It was divided into the external loop and internal loop subsystems.

The external loop subsystem is as follows

{\begin{cases} {\dot{x}}_{1} = f_{1} + g_{1} x_{2} \\ {\dot{x}}_{2} = f_{2} + g_{2} x_{3} \end{cases}

The internal loop subsystem is

{\dot{x}}_{3} = f_{3} + g_{3} u

By introducing the dynamic surface control to the external loop and L₁ adaptive control to the internal loop, the controller diagram is shown in Figure 1.

Figure 1.

Attitude subsystem control diagram.

According to the track angle instruction γ_d , the repeated virtual control is used to achieve the control effect with instruction filter introduced to solve the traditional problems of “calculation explosion.” The $x_{3 x}$ is the input variable in L₁ adaptive controller and the control law u can be obtained to control the height by tracking γ_d.

Backstepping control

Step 1

According to the equation ${\dot{x}}_{1} = f_{1} + g_{1} x_{2}$ , $e_{1} = x_{1} - x_{1 d}$ is defined, with the tracking error of track angle being

{\dot{e}}_{1} = f_{1} + g_{1} x_{2} - {\dot{x}}_{1 d}

$x_{2 c}$ is chosen as a virtual variable, and its design value is

x_{2 c} = [- k_{1} e_{1} + {\dot{x}}_{1 d} - f_{1}] / g_{1}

where k₁ is a positive design parameter.

There is the new signal $x_{2 d}$ and ${\dot{x}}_{2 d}$ which follow

ε_{2} {\dot{x}}_{2 d} + x_{2 d} = x_{2 c}

where $x_{2 d} (0) = x_{2 c} (0)$ .

Define $y_{2} = x_{2 d} - x_{2 c}$ , $e_{2} = x_{2} - x_{2 d}$ , equation (6) can be written as

{\dot{e}}_{1} = g_{1} (e_{2} + y_{2}) - {\dot{x}}_{1 d}

Step 2

The definition of $e_{2} = x_{2} - x_{2 d}$ is similar to that in step 1. The dynamic equation of pitching angle error e₂ is ${\dot{e}}_{2} = f_{2} + g_{2} x_{3} - {\dot{x}}_{2 d}$ .

Define $x_{3 c}$ as virtual parameter, there is

x_{3 c} = [- k_{2} e_{2} + {\dot{x}}_{2 d} - f_{2} - g_{1} e_{1}] / g_{2}

where k₂ is a positive design parameter. Equation (8) can be written as

{\dot{e}}_{1} = k_{2} e_{2} - g_{1} e_{1}

L₁ adaptive controller

The internal loop subsystem ${\dot{x}}_{3} = f_{3} + g_{3} u$ can be described in the linear time-varying system as

{\begin{cases} {\dot{x}}_{3} = A x_{3} + B (u + ρ x_{3} + σ) \\ y = c x_{3} \end{cases}

where A = 0, B = g₃, c = 1, and $u = δ_{e}$ is the control signal, with ρ and σ being the estimated time-varying parameters. Some assumptions are needed for the convenience of controller design.

Assumption 1

Unknown parameters ρ and σ are uniformly bounded. For known tight convex set Θ and Δ, there is $ρ (t) \in Θ$ , $σ (t) \in Δ$ , $\forall t \geq 0$ .

Assumption 2

Unknown parameters ρ and σ are continuous, differentiable, and uniformly bounded. There is $| \dot{ρ} (t) | \leq d_{ρ} < \infty$ and $| \dot{σ} (t) | \leq d_{σ} < \infty$ for $\forall t \geq 0$ , where $d_{ρ}$ and $d_{σ}$ are constant.

L₁ control law is composed of two parts

u = u_{1} + u_{2}

where $u_{1} = - K x_{3}$ is the static feedback part, K is the state feedback gain, and u₂ is adaptive control part to compensate the uncertainty. The internal loop subsystem can be described by

{\dot{x}}_{3} = A_{o} x_{3} + B b (u_{2} + ρ x_{3} + σ)

As the introduced state feedback control, the transfer function is as follows

H_{o} (s) = {(s I - A_{o})}^{- 1} B

In this article, the used state prediction model is

{\dot{\hat{x}}}_{3} = A_{o} {\hat{x}}_{3} + b (u_{2} + \hat{ρ} x_{3} + \hat{σ})

Define the state prediction error $e_{3} = {\hat{x}}_{3} - x_{3}$ , there is

{\dot{e}}_{3} = {\dot{\hat{x}}}_{3} - {\dot{x}}_{3} = A_{o} e_{3} + b (\tilde{ρ} x_{3} + \tilde{σ})

L₁ adaptive update law is in the following

\dot{\hat{ρ}} = Γ Pr o j (\hat{ρ}, - x_{3} e_{3} P b), \hat{ρ} (0) = {\hat{ρ}}_{0}

\dot{\hat{σ}} = Γ Pr o j (\hat{σ}, - e_{3} P b), \hat{σ} (0) = {\hat{σ}}_{0}

Estimate errors are $\tilde{ρ} = \hat{ρ} - ρ$ , $\tilde{σ} = \hat{σ} - σ$ and the dynamic equation is

{\begin{cases} \dot{\tilde{ρ}} = \dot{\hat{ρ}} - \dot{ρ} = Γ Pr o j (\hat{ρ}, - x_{3} e_{3} P b) - \dot{ρ} \\ \dot{\tilde{σ}} = \dot{\hat{σ}} - \dot{σ} = Γ Pr o j (\hat{σ}, - e_{3} P b) - \dot{σ} \end{cases}

where Γ > 0 and P > 0 is the solution of Lyapunov equation ${A_{o}}^{T} P + P A_{o} = 2 P A_{o} = - Q$ .

Then the adaptive controller can be designed as

u_{2} (s) = C (s) (\bar{r} (s) + k_{g} r (s))

where u₂(s), $\bar{r} (s)$ , and r(s) are the Laplace of u₂(t), $\bar{r} (t)$ , r(t), and $\bar{r} (t) = - ({\hat{ρ}}^{T} (t) x (t) + \hat{σ} (t))$ .

So, the L₁ controller diagram is shown in Figure 2 .

Figure 2.

L₁ adaptive controller diagram.

In all, the prediction model of loop system is in the following

\begin{array}{l} \dot{\hat{x}} (s) = \bar{G} (s) \bar{r} (s) + G (s) r (s) \\ \bar{G} (s) = H_{o} (s) (C (s) - 1) \\ G (s) = k_{g} H_{o} (s) C (s) \end{array}

It is necessary for the L₁ adaptive controller to satisfy the following conditions

λ = {‖ H_{o} (s) (C (s) - 1) ‖}_{L_{1}} ρ_{max} < 1

where $ρ_{max} = max_{θ \in Ω} \sum_{i = 1}^{n} | ρ_{i} |$ .

Velocity subsystem controller

According to $\dot{V} = f_{V} + g_{V} δ_{T}$ , the velocity track error is defined as $e_{V} = V - V_{r}$ , with its dynamic error equation

{\dot{e}}_{V} = f_{V} + g_{V} δ_{T} - {\dot{V}}_{r}

The expected controller is

δ_{T_{c}} = [- k_{V} e_{V} - f_{V}] / g_{V}

where k_V is the designed parameter and k_V > 0.

So, equation (23) can be written as

{\dot{e}}_{V} = - k_{V} e_{V}

Stability analysis

Proof 1

Select the Lyapunov function for the external loop subsystem in the following

L_{1} = \frac{1}{2} e_{1}^{2} + \frac{1}{2} y_{2}^{2}

L_{2} = \frac{1}{2} e_{2}^{2}

Taking the derivative of $L_{i} (i = 1, 2)$ , the following equations can be obtained

\begin{array}{l} {\dot{L}}_{1} = e_{1} {\dot{e}}_{1} + y_{2} {\dot{y}}_{2} \\ = - k_{1} e_{1}^{2} + g_{1} (e_{2} + y_{2}) e_{1} + y_{2} (- \frac{y_{2}}{ε_{2}} + M_{2}) \end{array}

{\dot{L}}_{2} = e_{2} {\dot{e}}_{2} = - k_{2} e_{2}^{2} - g_{1} e_{1} e_{2}

{\dot{L}}_{1} + {\dot{L}}_{2} = - k_{1} e_{1}^{2} + g_{1} e_{1} y_{2} - \frac{y_{2}^{2}}{ε_{2}} + y_{2} M_{2} - k_{2} e_{2}^{2}

$e_{1} y_{2} \leq \frac{1}{2} e_{1}^{2} + y_{2}^{2}$ and $y_{2} M_{2} \leq \frac{1}{2} y_{2}^{2} + \frac{1}{2} M_{2}^{2}$ , formula (30) can be rewritten as

\begin{array}{l} {\dot{L}}_{1} + {\dot{L}}_{2} \leq - (k_{1} - \frac{g_{1}}{2}) e_{1}^{2} - (\frac{1}{ε_{2}} - \frac{1}{2} - \frac{g_{1}}{2}) y_{2}^{2} \\ - (k_{2} - \frac{g_{2}}{2}) e_{2}^{2} + \frac{1}{2} M_{2}^{2} \end{array}

For

$k_{10} = k_{1} - \frac{g_{1}}{2} > 0$ , $\frac{1}{ε_{2}} - \frac{1}{2} - \frac{g_{1}}{2} > 0$ , $k_{20} = k_{2} - \frac{g_{2}}{2} > 0$ , the following equation can be obtained

{\dot{L}}_{1} + {\dot{L}}_{2} \leq - k_{10} e_{1}^{2} - k_{20} e_{2}^{2} + C

| e_{i} | \leq \sqrt{C / k_{i 0}}

The dynamic errors e₁ and e₂ are all bounded, so the external loop subsystem is stable.

Select the Lyapunov function of the internal loop subsystem in the following

L_{3} = P e_{3}^{2} + \frac{1}{Γ} {\tilde{ρ}}^{2} + \frac{1}{Γ} {\tilde{σ}}^{2}

Taking the derivative of (34), there is

\begin{array}{l} {\dot{L}}_{3} = 2 P e_{3} {\dot{e}}_{3} + \frac{2}{Γ} \tilde{ρ} \dot{\tilde{ρ}} + \frac{2}{Γ} \tilde{σ} \dot{\tilde{σ}} \\ = 2 P A_{o} e_{3}^{2} + 2 P b (\tilde{ρ} x_{3} e_{3} + \tilde{σ} e_{3}) + \frac{2}{Γ} \tilde{ρ} (\dot{\hat{ρ}} - \dot{ρ}) \\ + \frac{2}{Γ} \tilde{σ} (\dot{\hat{σ}} - \dot{σ}) \\ \leq - Q e_{3}^{2} - \frac{2}{Γ} \tilde{ρ} \dot{ρ} - \frac{2}{Γ} \tilde{σ} \dot{σ} \\ \leq - Q e_{3}^{2} + \frac{2}{Γ} | \tilde{ρ} \dot{ρ} + \tilde{σ} \dot{σ} | \end{array}

L = L_{1} + L_{2} + L_{3}

The projection operator guarantees that ρ ∈ Φ and σ ∈ Δ, and therefore the following formula is established

max_{t \geq 0} ({\tilde{ρ}}^{2} + {\tilde{σ}}^{2}) \leq 4 (max_{ρ \in Θ} ρ^{2} + max_{σ \in Δ} σ^{2})

Suppose $L_{3} \geq ρ_{m} / Γ$ , where

\begin{array}{l} ρ_{m} = \frac{4 P}{Q} (d_{ρ} max_{ρ \in Θ} | ρ | + d_{σ} max_{σ \in Δ} | σ |) \\ + 4 (max_{ρ \in Θ} ρ^{2} + max_{σ \in Δ} σ^{2}) \end{array}

\begin{array}{l} L_{3} = P e_{3}^{2} + \frac{1}{Γ} {\tilde{ρ}}^{2} + \frac{1}{Γ} {\tilde{σ}}^{2} \\ \geq \frac{4 P}{Q Γ} (d_{ρ} max_{ρ \in Θ} | ρ | + d_{σ} max_{σ \in Δ} | σ |) \\ + \frac{4}{Γ} (max_{ρ \in Θ} ρ^{2} + max_{σ \in Δ} σ^{2}) \end{array}

then equation (40) is obtained

P e_{3}^{2} \geq \frac{4 P}{Q Γ} (d_{ρ} max_{ρ \in Θ} | ρ | + d_{σ} max_{σ \in Δ} | σ |)

For P > 0, there is

Q e_{3}^{2} \geq \frac{4}{Γ} (d_{ρ} max_{ρ \in Θ} | ρ | + d_{σ} max_{σ \in Δ} | σ |)

From assumptions 1 and 2, the following equation can be obtained

| \tilde{ρ} \dot{ρ} + \tilde{σ} \dot{σ} | \leq 2 (d_{ρ} max_{ρ \in Θ} | ρ | + d_{σ} max_{σ \in Δ} | σ |)

Putting (41) and (42) into (35), there is

{\dot{L}}_{3} \leq 0

Suppose that $| e_{1} (0) | + | e_{2} (0) | \leq d_{c} < \infty$ , that d_c is bounded constant, that $y_{2} (0) = x_{2 d} (0) - x_{2 c} (0) = 0$ , and that $e_{3} (0) = 0$ , which yields

\begin{array}{l} L (0) = L_{1} (0) + L_{2} (0) + L_{3} (0) \\ = \frac{1}{2} e_{1}^{2} (0) + \frac{1}{2} y_{2}^{2} (0) + \frac{1}{2} e_{2}^{2} (0) + P e_{3}^{2} (0) + \frac{1}{Γ} {\tilde{ρ}}^{2} + \frac{1}{Γ} {\tilde{σ}}^{2} \\ \leq \frac{1}{2} d_{c}^{2} + \frac{4}{Γ} (max_{ρ \in Θ} ρ^{2} + max_{σ \in Δ} σ^{2}) \\ \leq \frac{1}{2} d_{c}^{2} + \frac{ρ_{m}}{Γ} \end{array}

Thus

L (t) \leq L (0) \leq \frac{1}{2} d_{c}^{2} + \frac{4}{Γ} (max_{ρ \in Θ} ρ^{2} + max_{σ \in Δ} σ^{2})

So the attitude subsystem is stable.

Proof 2

Select the following Lyapunov function

L_{V} = \frac{1}{2} e_{V}^{2}

Similar to proof 1, there is

{\dot{L}}_{V} = e_{V} {\dot{e}}_{V} = - k_{V} e_{V}^{2}

where the design parameter is

k_{V} > 0

For the Lyapunov function is negative, the velocity subsystem is stable.

Simulation verification and analysis

Using Matlab to calculate the balance state of different heights and velocities, this article simulates the situation of airdrop as shown in Table 6.

Table 6.

Different trim conditions for airdrop.

$V (ft \cdot s^{- 1})$	$H (ft)$	$α (°)$	$θ (°)$	$δ_{e} (°)$
221	16.404	1.9376	1.9376	0.0229
200	16.404	5.1535	5.1535	−1.0376
221	24.606	1.9433	1.9433	0.0172
200	24.606	5.1592	5.1592	−1.0376

The initial state before the airdrop is $V = 221 ft \cdot s^{- 1}$ , $α = 1.9376 °$ , $q = 0 ° / s$ , θ = 1.9376^°, and H = 16.404 ft. In the first 10 s, the plane is in the level situation, and at 10 s the goods start to move backward and are ready to leave the plane.

The L₁ adaptive backstepping controller for large transport aircraft airdrop heavy load is designed for the instability of the aircraft caused by the drop process. The controller parameters in the attitude subsystem are k₁ = 1.2 and k₂ = 4 with the static gains of K = −20, Γ = 1000, and Q = 1, and the low-pass filter is $C (s) = \frac{500}{s + 500}$ . The controller parameter of velocity subsystem is k_V = 1.2. The parameters of the PID control system are k_P = 0.8, k_I = 0.1, and k_D = 0.04.

Figure 3 is the chart for comparing the plane altitude and the velocity response with no controller or the L₁ adaptive backstepping controller. Figure 4 shows the dynamic response of flight angle, pitch angle, and pitch angle velocity in the two cases. From the comparison in Figures 3 and 4, we can see that the flight altitude, velocity, and attitude angles can be kept stable during the airdrop using this control method. If there is no proper control, the flight will change seriously, which will threaten the aircraft safety especially at the moment of cargos leaving the aircraft.

Figure 3.

Altitude and velocity response.

Figure 4.

Attitude angles response.

At the moment of airdrop, it is necessary to switch the control strategy for the model mutation and the switch moment is also very important for the flight’s robust performance.

Figures 5 to 9 show the control effect in the three cases: accurate switch, time delay of 1 s and 1 s ahead of switch time. It can be seen that the control strategy in this article obviously has the control performance with short overshoot, short response time, and short adjustment time, showing the good stability and dynamic performance. At last the rapidity of the control method is shown in Figure 10 in the certain control and environmental cases. From the simulation results, we can see that the controller based on the L₁ adaptive backstepping can guarantee the flight performance and high robustness.

Figure 5.

Attack angle dynamic response.

Figure 6.

Track angle dynamic response.

Figure 7.

Pitch angle dynamic response.

Figure 8.

Pitch rate dynamic response.

Figure 9.

Height dynamic response.

Figure 10.

Elevator response.

Conclusion

In this article, aimed at the airdrop, the backstepping is used for the velocity controller and the L₁ adaptive control combined with backstepping is proposed for the attitude control law. In view of the uncertainty existing in the system, the L₁ adaptive controller is used to estimate, compensate, and improve the control performance of the pure backstepping controller. By constructing the Lyapunov function, the system stability is proven. Simulation results show the effectiveness of the controller.

Footnotes

Acknowledgement

The authors would like to thank the anonymous reviewers for their helpful comments and valuable suggestions which improved this paper substantially.

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 study has been funded by the Shaanxi Industrial Science and Technology Key Project (grant no 2016GY-070) and Shaanxi Province Department of Education Key Project (grant no 2016JS017).

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