Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 787931, 12 pages

http://dx.doi.org/10.1155/2015/787931

## Adaptive Neural Control Based on High Order Integral Chained Differentiator for Morphing Aircraft

^{1}School of Automation, Northwestern Polytechnical University, Xi’an 710072, China^{2}Science and Technology on Aircraft Control Laboratory, AVIC Xi’an Flight Automatic Control Research Institute, Xi’an 710065, China

Received 28 July 2015; Accepted 17 September 2015

Academic Editor: Xinguang Zhang

Copyright © 2015 Zhonghua Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents an adaptive neural control for the longitudinal dynamics of a morphing aircraft. Based on the functional decomposition, it is reasonable to decompose the longitudinal dynamics into velocity and altitude subsystems. As for the velocity subsystem, the adaptive control is proposed via dynamic inversion method using neural network. To deal with input constraints, the additional compensation system is employed to help engine recover from input saturation rapidly. The highlight is that high order integral chained differentiator is used to estimate the newly defined variables and an adaptive neural controller is designed for the altitude subsystem where only one neural network is employed to approximate the lumped uncertain nonlinearity. The altitude subsystem controller is considerably simpler than the ones based on backstepping. It is proved using Lyapunov stability theory that the proposed control law can ensure that all the tracking error converges to an arbitrarily small neighborhood around zero. Numerical simulation study demonstrates the effectiveness of the proposed strategy, during the morphing process, in spite of some uncertain system nonlinearity.

#### 1. Introduction

With the development of morphing wing technology, the flight performance of an aircraft can be improved according to the current flight conditions [1–3]. The morphing aircraft are the flight vehicles that change their shape to either effectuate a change in mission or provide control authority for maneuvering [4, 5], without the use of discrete control surfaces or seams. Aircraft with morphing capability exhibit the distinct advantages of being able to fulfill multiple types of missions and to perform extreme maneuvers not possible with conventional aircraft [6, 7].

The field of morphing aircraft research is composed of a large array of interdisciplinary studies, including wing structure, actuation systems, aerodynamic modeling, nonrigid dynamics, and flight control [8]. A number of studies have focused on optimization of the actuator locations in the morphing structure units [9–11]. Other relative research work that involves the aeroelastics analysis is presented in [12]. The importance of the inertial forces and moments is studied in [13], with the goal of reducing the dynamics that must be dealt with in the flight control design. A methodology which is suitable for numerical calculation of the dynamic loads for a morphing aircraft is presented in [14]. In [15], linear parameter varying modeling is proposed for a folding wing morphing aircraft during the wing morphing process, whereas the longitudinal dynamic responses are numerically simulated based on the quasi-steady aerodynamic assumption.

Despite significant advances in the development of wing structure, actuation systems, and dynamic model, much work remains to be done to effectively control the morphing aircraft. The control system of a morphing aircraft must be capable of achieving consistent and robust performance meanwhile maintaining stability during large variations in the aircraft geometry, which may severely affect aerodynamic forces, moments of inertia, and center of mass.

For the disturbance rejection, a pair of linear controllers is synthesized for a linear input-varying morphing aircraft in [16]. A simple proportional state feedback control integrated with the eigenstructure assignment is proposed for the span-morphing aircraft in [17]. Based on a linear parameter varying model, self-gain scheduled controller is designed for the wing transition process in [18]. On the basis of varying linear parameter and classical methodology, a synthesized multiloop controller of a morphing unmanned aerial vehicle is formulated to guarantee a good performance subjected to large-scale geometrical shape changes in [19].

To cope with system uncertainties, adaptive control and neural network control techniques have been used for decades. For a linear morphing aircraft dynamic model, an indirect adaptive control method is designed in [6], which comprises the receding horizon optimal control law coupled with the modified sequential least squares parameter identification. In [20], a single network adaptive critic tracking controller design for a morphing aircraft is studied, wherein the set of initial weights of the neural network is determined by using a linear system model, which requires offline pretraining. Based on the concepts of feedback linearization, in [21], a combination of dynamic inversion and structured model reference adaptive control is used for the control of a morphing air vehicle. Typically a morphing aircraft exhibits highly nonlinear dynamics characteristics. Because of the morphing aircraft’s design and flight condition, it is extremely sensitive to change in physics as well as aerodynamic parameters. Almost all controller designs discussed above are based on linear models. Moreover the input saturation (physical limitation in engine) has not been considered in any work, which usually appears in many practical systems and severely degrades the closed-loop performance [22].

As a powerful nonlinear technique, backstepping control has been used for control system designs with strict-feedback form, extensively. With conventional backstepping, a possible issue is the explosion of complexity. This is caused by the repeating differentiations of certain nonlinear functions. To efficiently handle the system uncertainty in each subsystem, RBFNN with the universal approximation capability is employed in [23, 24]. Since RBFNN is used, we need to take derivatives of those radial basis functions, which will further lead to heavier calculation burden in each step design. Recently, the dynamic surface control was employed to solve this problem and many research results were presented [25, 26]. However, the determination of virtual control terms during the backstepping design requires tedious and complex analysis. More than one neural network is taken for approximation whose complexity increases like the order of the controlled backstepping design.

The motivation of this paper is to present a nonlinear robust adaptive neural controller for the morphing aircraft based on high order integral chained differentiator to achieve stability in the sweeping process where both system uncertainty and input restrictions are considered. The contribution of this paper can be summarized as follows.

Firstly, a nonlinear longitudinal model is derived from a curved-fitted model, with the center of mass position, aerodynamic forces, and the moments of inertia being varied with respect to the morphing parameters. The longitudinal model is then decomposed into altitude and velocity subsystems.

Secondly, the highlight is that the altitude subsystem dynamics is transformed into normal-feedback formulation and a robust adaptive neural controller using HICD is designed where only one neural network is employed to approximate the lumped uncertain system nonlinearity. The controller is considerably simpler than the ones based on backstepping which requires tedious and complex analysis for their virtual control terms. This feature guarantees that the computational burden of the algorithm can be reduced. Moreover the algorithm is convenient for real-time implementation on flight computers. Meanwhile, the adaptive control is proposed for velocity subsystem and an additional compensation system is employed to deal with input constraints, which will help engine recover from input saturation rapidly.

Finally, the Lyapunov synthesis based on stability analysis is used to prove that all the signals in the closed systems are semiglobally uniformly ultimately bounded with tracking error converging to a close neighborhood of origin.

The rest of the paper is organized as follows: Section 2 introduces the model of the morphing aircraft and formulates the normal output-feedback form of the altitude and velocity subsystems of longitudinal dynamics of the morphing aircraft. Section 3 briefly describes the background theory of RBFNN. Section 4 presents the adaptive neural controller design and the stability analysis for altitude and velocity subsystems. The simulation results are presented and discussed in Section 5. Section 6 gives the concluding remarks and future works.

#### 2. Problem Formulation

##### 2.1. Morphing Aircraft Model

The control-oriented model of the longitudinal dynamics of a morphing aircraft considered in this study is based on Seigler [4, 5]. This model comprises five state variables (, , , , and ) and two control inputs (, ), where is the velocity, is the altitude, is angle of attack, is the flight path angle (FPA), and is the pitch rate; and represent elevator deflection and thrust force, respectively. Considerwhere , , and represent drag force, lift force, and pitch moment, respectively; , , and denote the mass of aircraft, moment of inertia about pitch axis, and gravity constant; , , , and represent inertial force and moment caused by morphing process; is the position of engine in the body axis; denotes the static moment distributed in the body axis of ; the related definitions are given as follows:where represents the sweep angle, denotes the air density, is the wing surface, represents the mean aerodynamic chord, and is the wingspan. and denote the dynamic pressure and pitch moment. , , and are the total aerodynamic lift force coefficient, drag force coefficient, and pitching moment coefficient, respectively. and represent the mass of aircraft’s wing and body. and denote the position of aircraft’s wing and body in the aircraft-body coordinate frame.

We assume that the engine model can be expressed as follows [27].

*(A) Engine Rate*. The dynamics for the engine speed is modeled by a first-order linear system with the time constant and the engine speed reference signal as follows:

*(B) Thrust Force*. The thrust force is generated by the propeller and can be expressed with dimensionless coefficients. The dimensionless thrust coefficient iswith the ratio , where the diameter of the propeller is , the engine speed is , and the airspeed is . Here we assume that is equal to . The thrust force is computed as shown below:

*Remark 1. *It is important to point out that , , , , , , , , and are associated with sweep angle in the morphing process. Their functional relationships will be shown later in Section 5.

##### 2.2. System Transformation

*(A) Altitude Subsystem*. The tracking error of the altitude is defined as . Furthermore, the altitude command is transformed into the desired flight path angle (FPA). The demand of flight path angle is generated as [22]If and are chosen appropriately and the FPA is controlled to follow , then the altitude error is regulated to zero exponentially.

*Remark 2. *Since the control problem considered in this paper only takes into account cruise trajectories and does not consider the aggressive maneuvering, the thrust can be neglected since it is generally much smaller than the lift. In order to transform the altitude subsystem into strict-feedback form, in (3) is regarded as an unmodeled term.

Define , , , , , ; the strict-feedback forms of equations of the altitude (3)–(5) are rewritten as where

*Assumption 3. *, , , , , and are unknown smooth functions; we assume that there exist positive constants , , , and such that , , . There also exist constants and such that , . Meanwhile, in this paper, we assume that all the system states can be measured and there is no time-delay in the signal transmission.

Lemma 4 (high order integral chained differentiator [28]). *Suppose the function and its first derivatives are bounded. Consider the following linear system:where is a small positive constant and parameters to are chosen such that the polynomial is Hurwitz. Then*

In the following, we show that original system (12) can be transformed into the normal form with respect to the newly defined state variables. Let and . The derivative of with respect to time is formulated aswhere , .

Similarly, let and its time derivative is induced by where and .

As a result, strict-feedback system (12) can be described as the following normal output form with respect to the newly defined state variables , , and :

*(B) Velocity Subsystem*. With the modeling uncertainties and external disturbance existing, the uncertain nonlinear model can be formulated aswhere , , . is the nominal parts of ; is the unknown system uncertainties of ; is the external disturbance and is the lump of system uncertainty.

*Remark 5. *It should be noted that , are totally unknown and need to be approached by NN in the subsequent developments. For the newly defined states , , and , an HICD will be introduced to estimate them. From Assumption 3, it is also noted that there exist constants and such that and .

#### 3. Neural Networks

In many references of robust adaptive control of uncertain nonlinear systems, the RBFNNs are usually employed as approximate model terms for the unknown nonlinear and continuous function terms using their inherent approximation capabilities [25]. As a class of linearly parameterized NNs, RBFNNs are adopted to approximate the unknown and continuous function which can be written as follows:where is an input vector of NN, is a weight vector of the NN, is a basis function, is the approximation error which satisfies , and is a bounded unknown parameter.

In general, an RBFNN can smoothly approximate any continuous function over the compact to any arbitrary accuracy aswhere is the optimal weight value and is the smallest approximation error. The Gaussian basis function is written in the form of where and are the center and width of the neural cell of the th hidden layer.

*Remark 6. *There exists an RBFNN in the form of (21) and an optimal parameter vector such that . denotes the supremum of the reconstruction error that is inevitably generated. In what follows, the estimation of is denoted as .

#### 4. Control Design and Stability Analysis

It is easy to note that is mainly related to and is mainly affected by . Therefore, the dynamics can be decoupled into altitude and velocity subsystem and we design the altitude and velocity controller separately. The structure of the proposed control scheme is presented in Figure 1.