Hewei Zhao

（Detection Guidance and Control Teaching Research Section, Naval Aviation University, Yantai 264001, Shandong, China）

Abstract: This paper presents an integrated guidance and control model for a flexible hypersonic vehicle with terminal angular constraints. The integrated guidance and control model is bounded and the dead?zone input nonlinearity is considered in the system dynamics. The line of sight angle,line of sight angle rate, attack angle and pitch rate are involved in the integrated guidance and con?trol system. The controller is designed with a backstepping method, in which a first order filter is employed to avoid the differential explosion. The full tuned radial basis function(RBF) neural net?work(NN) is used to approximate the system dynamics with robust item coping with the recon?struction errors, the exactitude model requirement is reduced in the controller design. In the last step of backstepping method design, the adaptive control with Nussbaum function is used for the unknown dynamics with a time?varying control gain function. The uniform ultimate boundedness stability of the control system is proved. The simulation results validate the effectiveness of the controller design.

Key words: hypersonic vehicle；terminal angular constraint；dead?zone input nonlinearity；full tuned radial basis function(RBF) neural network(NN)；integrated guidance and control

Promising military application values, and cost?effective technology of hypersonic vehicle are concerned by many researchers both home and abroad. However, many challenges of control sys?tem design for hypersonic vehicle stemming from highly coupled and nonlinear characteristics of its dynamic behavior are needed to solve with some advanced methods. In many studies, the longit?udinal dynamic models are investigated.

The T?S model is described and used to design a controller with some intelligent methods[1]. Currently, nonlinear control and intel?ligent control are integrated to design controllers[2], fuzzy control, adaptive control, and dynamic surface control are employed to design controllers for hypersonic vehicles. Also in Ref. [3],the controller is designed with adaptive and slid?ing mode control methods. Different from the traditional flight vehicle, flexible issue re?searched in many papers is important to solve for hypersonic vehicle. A coupling?observer?based nonlinear control is designed for flexible air?breathing hypersonic vehicles[4]. Robust and ad?aptive methods are employed to design the con?troller for flexible hypersonic vehicles[5].

Currently, neural control is employed to design the controller for hypersonic vehicle in many studies. Nonlinear functions of hypersonic vehicle dynamics are approximated by neural network(NN) with updating laws designed for the estimations of neural network weight vectors and construction errors[6]. A novel adaptive neur?al controller is designed for a constrained flexible hypersonic vehicle and by employing a minimal?learning parameter method to estimate the norm of ideal weight vectors rather than their ele?ments, there are only two adaptive parameters required for neural approximation[7]. Thus, the computational burden is lower than the ones de?rived from neural back?stepping schemes. In nu?merous studies, radial basis function(RBF) NN is used for the controller design of hypersonic vehicle, the updating law is introduced only for weight vector with considering the learning speed of NN, and however, the approximation ability to unknown nonlinear functions is as important as learning speed for NN. For RBF NN, the updat?ing laws should be designed for weight vector,central point of radial basis function, and sphere influence of radial basis function.

For large flight envelope of hypersonic vehicle, the controller should be designed for fast changing states. Also the guidance scheme and the flight control system need to be highly integ?rated in order to provide a robust stable high performance flight. The guidance and attitude system controller is designed in the dive phase[8].The integrated guidance and control with a ter?minal angular constraint is studied for improv?ing the hit accuracy[9?10]. Furthermore, the input nonlinearity is not considered in the procedure of integrated guidance and control design[11?17]. The presence of dead?zone input nonlinearity in feed?back control systems may cause severe deteriora?tion of the system performance, and due to the input nonlinearity, the control gain function is time varying. In this study, the dead?zone input nonlinearity will be discussed during the integ?rated guidance and control design for the hyper?sonic vehicle. A new prescribed performance con?trol method is employed to design the controller guaranteeing the transient performance of sys?tem[18?26]. Also the prescribed performance meth?od is used for the controller design of hypersonic vehicle. The prescribed performance fine attitude controller is designed for flexible hypersonic vehicle with considering unknown initial errors[27].

This paper is organized as follows. The in?tegrated guidance and control model dynamics of a flexible hypersonic vehicle and problem formu?lation are described in Section 1. In Sections 2 and 3, an integrated guidance and control sys?tem is designed and the system stability is proved separately. Simulation is carried out and the results are concluded in Section 4. Mean?while, some conclusions are presented in Section 5.

1 Model Dynamics and Problem Formulation

1.1 Hypersonic vehicle dynamics with flexible states

The detail of the parameter values can be found in Ref. [28].

1.2 Relative motion equations between the hypersonic vehicle and the target

The relative motion between the hypersonic vehicle and the target is needed to obtain for es?tablishing the integrated model. The relative mo?tion information is shown in Fig. 1.

Fig. 1 Relative motion between hypersonic vehicle and target

1.3 Analysis of terminal angular constraint

1.4 Integrated guidance and control model with terminal angular constraint

Many papers do not consider the actuator dynamics of elevator defection, the input nonlin?earity described in Ref.[6] is shown in Fig. 2 with the following description:

Then Eq.(35) can be described as

Fig. 2 Nonlinear input function

According to the conclusion above, the design is on the control of rigid body. Based on the functional decomposition, the dynamics can be divided into a velocity subsystem and an alti?tude subsystem. In this study the controller is designed for the altitude subsystem only.

Remark 2 According to the timescale conclu?sion in Ref. [29], the velocity is considered as slow dynamics. In the progress of control design for the altitude subsystem, the velocity is con?sidered as constant.

For the altitude subsystem, the tracking er?ror is defined as

1.5 Control goal

Fig. 3 Control scheme

1.6 Preliminaries

2 Controller Design for Integrated Model

3 Stability Analysis

4 Simulation

Fig. 4 Control input u and elevator deflection

Fig. 5 Line of sight angle

Fig. 6 Line of sight angle rate

Fig. 7 Pitch rate

Fig. 8 Attack angle

Fig. 9 Altitude

Fig. 10 Flight path angle

Fig. 11 Flexible states

Fig. 12 Attack angle tracking error

Fig. 13 Pitch rate tracking error

5 Conclusions

This paper presents the integrated guidance and control model for the longitudinal dynamics of a flexible hypersonic vehicle considering the dead?zone input nonlinearity, in which the ter?minal angular constraint can be guaranteed.

① The backstepping method is employed to design the controller. The first order filters are used to calculate the differential of ideal values of system states and the differential explosion from the backstepping method is avoided.

②The full tuned RBF NN is employed to approximate the system unknown nonlinear dy?namics, thus the requirement of exactitude mod?el is reduced in the controller design process.

③ In the controller design, a terminal angu?lar constraint and the dead?zone input nonlinear?ity issue are considered.

④ The stability of control system is guaran?teed via the Lyapunov approach. The simulation results can demonstrate the effectiveness of the proposed control method.