Nonlinear robust adaptive NN control for variable-sweep aircraft

1. Introduction

The morphing and bio-inspired UAV designs has attracted extensive focus because of the ability to optimize flight in different conditions via altering shape [1], possess the remarkable performance, e.g. flight in multiple mission types and extreme maneuver performance, of which is not possible for a traditional fixed-wing aircraft [2, 3]. However, due to the significant changes in mass distribution and aerodynamic parameters while altering shape, an uncertain, time-varying and complicated nonlinear dynamic model is posed which makes a crucial importance in the transient dynamic characteristics [2-4]. And the characteristics induced by configuration change are of paramount importance in controller design for a morphing aircraft [3]. So, the issue of controller design for morphing aircraft has been attached more importance by most researchers.

In order to overcome the controller design problem mentioned above, several linear controllers have been designed. In [5], a total of 350 operating points was computed by considering a generic morphing UAV mission profile to generate LPV model, and then a set of inner-loop gains to provide stability whereas the outer-loop LPV controller guarantees global quadratic stability was proposed based on the multiloop topology. Similarly, self-scheduled controller is used based on LPV model in [6] and a smooth switching controller based on linear parameter variable (LPV) model is applied to control morphing aircraft in [7]. In addition, the dynamic equations are converted into switched system in several special points in [8], and then a finite-time boundedness controller is proposed to guarantee steady flight in the morphing process. Furthermore, in [9], the time varying characteristic equation based on the linear time-invariant characteristic equation and the concept of time varying pole are posed to analysis the time-varying dynamic mode of variable-sweep morphing. In [10], an active disturbance rejection controller is proposed based on ADRC without LPV or switched model while the un-modeled dynamics and aerodynamics are observed and compensated as a generalized disturbance in real time.

The controller design methods above are at the basis of linear control with linearized model or linear input variable model. Those linear methods highly depend on the prior accurate aerodynamic parameters, which are strict for morphing controller design. In view of the limitation of a linear controller, nonlinear controllers (e.g. high-order integral chained differentiator control and first-order sliding mode differentiator control) are proposed to overcome the problem in [11, 12], but FPA is limited to $\mathrm{s}\mathrm{i}\mathrm{n}\gamma \approx \gamma$ and the errors of differentiator have not been considered. Therefore, controller for a morphing aircraft with the ability to solve the problem leaded by time-varying uncertain model should be further investigated in details.

To design a nonlinear controller with taking more factors into account, a nonlinear neural network (NN)-based dynamic surface control method is proposed to solve the problem induced by transient dynamic of variable-sweep aircraft with model uncertainties. The control objective is to maintain altitude and velocity during the wing transition process where significant changes in mass distribution, aerodynamic forces and moments occur. On the basis of functional decomposed altitude subsystem and velocity subsystem, a synthesis controller is posted with backstepping method. Compared with traditional flight control coping strict feedback form, the proposed controller is designed based on non-affine pure feedback form. Filtered signal is used to circumvent the algebraic loop problem in altitude controller design to remove the constraint on FPA. In order to overcome the problem of ‘explosion of complexity’ inherent in the conventional back-stepping method, the dynamic surface control(DSC) is utilized to compute the derivatives of virtual control laws and minimal learning parameters(MLP) techniques is employed to regulate the norm of NN’s weight vector to decrease the online updated learning parameters. According to the Lyapunov-LaSalle invariant set theorem, it is proved that the proposed controller is semi-globally uniformly ultimately bounded, of which tracking errors converge to an arbitrary small neighborhood of the origin via choosing designed constants appropriately. Case studies illustrate that the proposed controller shows high morphing flight performance and good feasibility.

2. Aerodynamic parameters

The schematic illustration of the variable-sweep aircraft used for calculating aerodynamic parameters is presented in Fig. 1. The inspiration of the configuration comes from our previous work in [13]. The length of the body is 5.33 m with a maxmum 5.893 m of span and the total mass is assumed to be 200 kg. Datcom is employed for aerodynamics calculation which was developed by the United States Air Force (USAF) for engineering estimation of the aircraft aerodynamics. The program calculates aerodynamics based on experimental results previously which is quite practical to quickly and economically estimate the aerodynamics of a wide variety of aircraft configuration designs [14]. Compared with wind tunnel experiments and computational fluid dynamic (CFD) which are expensive and time consuming, the advent of Datcom has provided variable-sweep aircraft designers with a relatively rapid and economical tool that provides a basic approximation of the aerodynamics of the aircraft.

Fig. 1Schematic illustration of the aircraft

In this paper, Datcom is employed to calculate these derivatives for the aircraft when symmetric planform changes are implemented for that we focus on the longitudinal altitude controller design. To reduce the difficulties in calculating aerodynamic parameters at the basis of guarantee accuracy, the following assumptions are made which are commonly used [1, 15]:

Assumption 1: The outboard wing shape is retentive through an assumed mechanism. That is to say the outboard wing shape is parallel to the axis of the fuselage during wing transition process.

Assumption 2: For simplicity, it is desirable to maintain the quasi-steady approximation; hence, aerodynamic derivatives will be determined for a specific aircraft design.

Assumption 3: The area of the wings is constant in the wing transition process as well as when the sweep angle is 0°.

Angle of attack and sweep angle have a great influence in the aerodynamic coefficients. In addition, considering that the Mach and height are changed in a small region for the goal of this paper being to maintain altitude and velocity during variable sweep process, the velocity and altitude are set to be constant during calculating the aerodynamic coefficients. At those basics, the lift coefficient, drag coefficient and pitching moment coefficient calculated by DATCOM are shown in Figs. 2-4.

Fig. 2Lift coefficient

Fig. 3Drag coefficient

Fig. 4Pitching moment coefficient

3. Dynamic model of the variable-sweep aircraft

In this paper, we concentrate on guaranteeing cruise state to be constant or fluctuate in a small region during the variable sweep aircraft sweeps back. Dynamic model considered here is based on [16]:

where:

4. Altitude controller design

To simplify the controller design process, the velocity controller and the altitude controller are designed separately as in [17, 18]. In this section, a nonlinear robust adaptive NN altitude controller based on backstepping method is proposed for the variable-sweep aircraft. MLP and DSC techniques are used to cope with the problems of ‘explosion of complexity’ as well as the online updated parameters being too much.

Remark 1: In order to transform the altitude system into pure-feedback system, $F_{Ikz}$ is regarded as an un-modeled term as in ref. [10].

Remark 2: Considering that the mass of the wing is much smaller than that of the aircraft, it is reasonable to suppose that the moment of inertia $I_y$ is a known constant during the wing transition process for the variable-sweep aircraft.

Define that $x_1=h$, $x_2=\gamma$, $x_3=\theta$, and $x_4=q$, the dynamic model can be expressed in the non-affine pure feedback form, which is expressed as follow:

where:

Lemma 1 [19]: Lyapunov-LaSalle invariant set theory: for a differential dynamic system $\dot{x}=f(x,t)$ where $L:D\to R_{+}$ is a continuous and differentiable function. $\mathrm{\Omega }=\left\{x\left|L\left(x\right)\le p\right.\right\}$ is a compact set where $p\in R_{+}$ and $\mathrm{\Omega }\subset D$, $E=\{x\in \mathrm{\Omega }\left|L\left(x\right)=p\right.\}$. If $\forall x\in E$, then $\dot{L}\left(x\right)\le$ 0, we call $\mathrm{\Omega }$ a invariant set of the system, and get: $x\left(t_0\right)\in \mathrm{\Omega }\Rightarrow x\left(t\right)\in \mathrm{\Omega }$, $\forall t\ge t_0$.

Lemma 2 [20]: the hyperbolic tangent function is continuous and differentiable, and the following inequalities are established $\forall q\in R$ and $\forall c>$ 0:

Based on backstepping method and DSC, the tracking errors are defined as below:

where $e_1$ is the system tracking error, ${\alpha }_i$ is the virtual control, $e_i$ is the tracking errors of virtual controls and ${\alpha }_{if}$ is the low-pass filtered signal of ${\alpha }_{i-1}$.

The control scheme is developed in the framework of back-stepping technique, which contains four-step recursive design procedure.

Step 1: Define $e_1=x_1-y_d$, and then ${\dot{e}}_1={\dot{x}}_1-{\dot{y}}_d$.

Define the following Lyapunov function $L_{e_1}=0.5e_1^2$, take derivation of $L_{e_1}$ and then:

In this paper, $f_1(x_1,x_2)$is an unknown function where neural network can be employed to approximate the function. Filtered signal is used to circumvent the algebraic loop problem in the backstepping method [21]. $x_{2f}$ is a filtered signal of $x_2$ defined as: $x_{2f}\equiv H_L\left(s\right)x_2\approx x_2$. The filtered signal error $\mathrm{\Delta }{f}_1$ and the minimum NN approximation error ${\epsilon }_1$ are bounded that $\left|{\epsilon }_1+\mathrm{\Delta }{f}_1+d_1\right|\le {\delta }_1^{\mathrm{*}}$.

In Eq. (8), the following inequality is obtained at the basis of Young inequality:

where $a_1$ is a positive constant needed to be designed.

Take ${\delta }_1^{\mathrm{*}}$ and Eq. (9) into Eq. (8), we can get:

To make the error $e_1$ as small as possible, the virtual control ${\alpha }_1$ and updating laws are chosen as:

where ${\tilde{\theta }}_1={\theta }_1-{\widehat{\theta }}_1$, ${\tilde{\delta }}_1={\delta }_1^{\mathrm{*}}-{\widehat{\delta }}_1$, ${\theta }_1={‖W_1^{\mathrm{*}}‖}^2$, and ${\widehat{\theta }}_1$ is the estimation of ${\theta }_1$.

Augment $L_{e_1}$ to obtain:

where ${\gamma }_1$ and ${\beta }_1$ are positive constants.

Take derivation of Eq. (12) along Eq. (11) and consider Lemma 2 to obtain:

Step 2: Define $e_2=x_2-{\alpha }_{2f}$ and then ${\dot{e}}_2={\dot{x}}_2-{\dot{\alpha }}_{2f}$.

Define the following Lyapunov function $L_{e_2}=0.5e_2^2$, take derivation of $L_{e_2}$ and then:

The filtered signal error $\mathrm{\Delta }{f}_2$ and the minimum NN approximation error ${\epsilon }_2$ are bounded that $\left|{\epsilon }_2+\mathrm{\Delta }{f}_2+d_2\right|\le {\delta }_2^{\mathrm{*}}$.

In Eq. (12), the following inequality is obtained at the basis of Young inequality:

where $a_2$ is a positive constant needed to be designed.

Take ${\delta }_2^{\mathrm{*}}$ and Eq. (15) into Eq. (14), we can get:

To make the error $e_2$ as small as possible, the virtual control ${\alpha }_2$ and updating laws are chosen as:

where ${\tilde{\theta }}_2={\theta }_2-{\widehat{\theta }}_2$, ${\tilde{\delta }}_2={\delta }_2^{\mathrm{*}}-{\widehat{\delta }}_2$, ${\theta }_2={‖W_2^{\mathrm{*}}‖}^2$, and ${\widehat{\theta }}_2$ is the estimation of ${\theta }_2$.

Augment $L_{e_2}$ to obtain:

where ${\gamma }_2$ and ${\beta }_2$ are positive constants.

Take derivation of Eq. (18) along Eq. (17) and considering Lemma 2 to obtain:

Step 3: Define $e_3=x_3-{\alpha }_{3f}$, and then ${\dot{e}}_3={\dot{x}}_3-{\dot{\alpha }}_{3f}$.

Similarly, to make the error $e_3$ as small as possible, the virtual control ${\alpha }_3$ is chosen as:

where $k_3$, ${\gamma }_3$, $v_3$ are positive constants to be designed.

Define the following Lyapunov function $L_3=0.5e_3^2+{\tilde{\delta }}_3^2/2{\gamma }_3$, and take derivation of $L_3$ along Eq. (20), we can get:

Step 4: Define $e_4=x_4-{\alpha }_{4f}$, and then ${\dot{e}}_4={\dot{x}}_4-{\dot{\alpha }}_{4f}$.

According to step 2, define $L_{e4}=0.5e_4^2$, we can get:

The moment of inertia $I_y$ varies with sweep angle which leads to the uncertainty of $C_{{\delta }_e}$. And to overcome the uncertainty, we use ${\widehat{C}}_{{\delta }_e}$ to instead of $C_{{\delta }_e}$. To make the error $e_4$ as small as possible, the control input ${\delta }_e$ and the adaptive law are taken as:

Augment $L_{e4}$ to obtain:

where $k_1=$ 5, $k_1=$ 10, $k_1=$ 5, and ${\beta }_4$ are positive constants.

Considering Lemma 2, the time derivative of $L_4$ along Eq. (23) is given:

Theorem 1. Consider the longitudinal altitude system described by Eq. (6), under the assumptions of remark 1, remark 2 and the initial condition $x_1\left(0\right)\le \mathrm{\Omega }$, use the control law Eq. (23), filtered signal Eq. (7), virtual control and updating laws in Eq. (11), (17) and (20), then the closed-loop signal are semi-globally bounded, forcing the altitude tracking error to an arbitrary small neighborhood of the origin.

Proof: Consider the following Lyapunov function candidate:

Along Eq. (13), (19), (21) and (25), the time derivative of $L$ is obtained:

From Eq. (7), we obtain:

and:

Define:

where $B_{i+1}(\cdot )$ is an unknown continuous function.

And then:

Consider that:

From Young inequality, we get:

Take Eq. (28)-(33) into Eq. (27), we get:

Then:

where $C_1$ is positive and $C_2$, ${\omega }_1$, ${\omega }_2$ are positive constant which are designed as below:

And then inequality Eq. (35) satisfies:

Define a compact set $\mathrm{\Omega }=\left\{x\left|L\left(x\right)\le p\right.\right\}$ and $E=\{x\in \mathrm{\Omega }\left|L\left(x\right)=p\right.\}$. And display $C_3$ by $C_3\le p$, and so if $L\left(x\right)\le p$, then $\dot{L}\left(x\right)\le 0$. $\mathrm{\Omega }$ is a compact set of $x\left(t\right)$. The unknown continuous function $B_{i+1}(\cdot )$ is bounded in the compact $\mathrm{\Omega }$. From Lemma 1, it is proved that the proposed controller is to be semi-globally uniformly ultimately bounded.

From Eq. (37), it is known that:

when $t\to +\mathrm{\infty }$, we get:

The tracking error is bounded by $C_3$which can be adjusted to arbitrary small of zero. So, the proposed controller can keep the altitude of the variable-sweep aircraft during the wing transition process.

5. Velocity controller design

In this section, a velocity controller is proposed based on NN adaptive control method. We first define:

First, we define $e_v=V-V_d$, and then:

In this paper, $f_v(x_v,T)$ is an unknown function, and an adaptive NN is used to approximate. $T_f$ is the filtered signal of $T$ defined as:

where $H_L\left(s\right)$is a Butterworth low-pass filter. The order of the Butterworth LPF utilized is 2 and the corresponding filter parameters of Butterworth filter is cited from [22].

Define the following Lyapunov function $L_v=0.5e_v^2+{\tilde{\delta }}_v^2/2{\gamma }_v+{\tilde{\theta }}_v^2/2{\beta }_v$ where ${\gamma }_v$ and ${\beta }_v$ are positive constants and $\left|{\epsilon }_v+\Delta f_v+d\right|\le {\delta }_v^{\mathrm{*}}$.

To make the error $e_v$ as small as possible, the control $T$ and updating laws are chosen as: