Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 591789, 10 pages
http://dx.doi.org/10.1155/2015/591789

Research Article

Neural Network-Based Adaptive Backstepping Control for Hypersonic Flight Vehicles with Prescribed Tracking Performance

Zhu Guoqiang and Liu Jinkun

School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

Received 1 December 2014; Revised 11 April 2015; Accepted 15 April 2015

Academic Editor: Hak-Keung Lam

Copyright © 2015 Zhu Guoqiang and Liu Jinkun. 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.

An adaptive neural control scheme is proposed for a class of generic hypersonic flight vehicles. The main advantages of the proposed scheme include the following: (1) a new constraint variable is defined to generate the virtual control that forces the tracking error to fall within prescribed boundaries; (2) RBF NNs are employed to compensate for complex and uncertain terms to solve the problem of controller complexity; (3) only one parameter needs to be updated online at each design step, which significantly reduces the computational burden. It is proved that all signals of the closed-loop system are uniformly ultimately bounded. Simulation results are presented to illustrate the effectiveness of the proposed scheme.

1. Introduction

During the past decades, hypersonic flight vehicles (HFVs) have received a great deal of attention. They may represent more cost-efficient and reliable access to space routine and are especially suitable for prompt global response, as well as offering worldwide air superiority because of the high speed and endurance [1–5]. In this paper a nonlinear generic model of HFVs is adopted, which has been widely used by various researchers [6–8]. The dynamics of HFVs are highly nonlinear with strong couplings between the propulsive and aerodynamic effects. The requirements of flight stability and high speed response make the onboard flight control of HFVs quite difficult [9, 10]. Besides, modeling inaccuracy can result in strong adverse effects on the performance of HFVs control systems. Thus, the controller design for HFVs is challenging and must guarantee closed-loop stability and desired performance [11].

Recently, feedback control strategy based on nonlinear control theory has been used for HFVs, such as sliding mode control [3], minimax linear quadratic regulator control [12, 13], genetic algorithm [14], and sequential loop closure controller design [15]. In [16], the adaptive backstepping method was used to design controller for the HFVs model, while fuzzy logic and neural networks were used to approximate the unknown system dynamics in [17–19]. Adaptive dynamic surface control schemes were proposed by [20, 21] to avoid the derivatives of nonlinear functions. The nonlinear dynamic inversion method was used to design a robust controller. In [3, 14], feedback linearization techniques were applied to design nonlinear controllers for the longitudinal motion of a hypersonic aircraft containing aerodynamic uncertain parameters. This approach leads to a complicated high-order Lie derivatives and is hard to perform a robustness analysis when considering uncertainties. In [22], a neural network controller for a nonlinear flight dynamic system was designed by using the adaptation mechanism to deal with the effects of aerodynamic modeling errors.

In the control design for HFVs, an important issue is tracking performance. Traditionally, the controller for HFVs guarantees the tracking error convergence to a residual set. Moreover, the transient behavior such as overshoot, undershoot, and convergence rate are difficult to be established analytically. In [23–25], a prescribed performance scheme is proposed for one-class nonlinear systems; this approach is to construct a prescribed performance function that converts the tracking error into a new variable. Therefore the tracking performance can be characterized by a prescribed constraint function. Besides, the prescribed performance approach with new definition is applied in a class of uncertain strict-feedback systems [26], strict-feedback time-delay systems [27], and MIMO systems [28], respectively.

A drawback of adaptive NNs [22] or FLSs [29, 30] schemes is that the number of adaptation laws generally depends on the neural network nodes or the fuzzy rules. That is, with an increase of the nodes or the rules, the parameters to be estimated may be greatly increased. To solve this problem, we propose a new method by estimating the norm of the NNs weights rather than estimating every item of the weight vector [31–33].

In this paper, we separate the longitudinal model of HFVs into two parts: the velocity subsystem and the altitude subsystem. Velocity and altitude controllers are designed separately. For the velocity subsystem, a dynamic inversion controller with radial basis function neural networks (RBF NNs) is proposed to track a desired velocity trajectory. The altitude subsystem is transformed into a strict-feedback form. Then an adaptive backstepping controller is designed to track a desired altitude trajectory. The main contribution of this paper is described as follows:

(1)We introduce a performance function, and a new error constraint variable is used as a virtual tracking error variable to ensure the prescribed transient performance. By extending the prescribed tracking performance technique proposed in [23, 24] to HFVs, it is shown that the tracking errors can converge to predefined arbitrarily small residue sets with prescribed convergence rate and maximum overshoot.(2)RBF NNs are employed to compensate for complex and uncertain terms to solve the problem of controller complexity. By using the minimal learning technique [31–33], only one parameter needs to be updated online at each design step regardless of the NNs input-output dimension and the number of NNs nodes. As a result, the number of adaptation laws, which generally depends on the neural network nodes, and the computational burden are greatly reduced.(3)With the bounded of the virtual control gain , the singularity problem by the estimation of is avoided without any effort, and both low and up bounded will not appear in the control law and will be used only for analysis; they can be unknown.

The rest of this paper is organized as follows. In Section 2, the nonlinear longitudinal dynamic model of HFVs is presented. The controllers design and the stability analysis are given in Section 3. The simulation results are illustrated in Section 4, followed by conclusions of this paper in Section 5.

2. Problem Formulation and Preliminaries

2.1. Longitudinal Model of HFVs

The model considered in this paper is taken from the NASA Langley Research Center [2, 3]. Cruising at a Mach number of 15 and at an altitude of 110000 ft, the longitudinal hypersonic flight model is given by

where is the velocity, the flight path angle, the altitude, the attack angle, the pitch rate, the elevator deflection, and the throttle setting. , , , and represent the thrust, drag, lift-force, and pitching moment, respectively, which can be expressed as

with

The nominal values of inertial and aerodynamic parameters are given in Table 1. Besides, at trimmed cruise condition, ft/s, ft,  rad, , and  rad.

Table 1: Parameters of the HFV.

Symbol
Parameter
Value

Mean aerodynamic chord
80 ft

Moment of inertia
slug·ft²

Reference area
3603 ft²

Air density
slugs·ft⁻³

Mass of aircraft
9375 slug

Radius of the earth
20903500 ft

Gravitational constant
 ft³/s²

The engine dynamics can be modeled by a second-order system:

Therefore, by selecting the commanded value as the new control input, the HFV is composed of five state variables and two control inputs , while the outputs to be controlled are selected as . The design objective is such that the outputs track the desired altitude and velocity commands with prescribed tracking performance.

From (1), it can be inferred that the main contribution in the change of flight vehicle velocity is from the throttle setting . The altitude change is related mainly to the elevator deflection . Thus, it is reasonable to divide the system into two loops: the velocity loop and the altitude loop.

Note that the thrust term is generally much smaller than the lift , velocity is high, and the flight path angle is typically very small during the trimmed cruise condition, which justify the following approximation.

Assumption 1 (see [7, 19]). The thrust term , and the term .
Defining that denotes the pitch angle, we have . Then, we define state variables as , with , , , and . For simplicity, let , so the altitude subsystem can be written as

where

Since the values of the inertial and the aerodynamic parameters are uncertain, the aforementioned and , , are unknown smooth functions. Moreover, it is easy to check that are always strictly positive. With these observations in mind, we have the following assumption.

Assumption 2. There exist positive constants and such that .

Remark 3. It is worth noting that, in the proposed scheme, both and will not appear in the control law and will be used only for analysis; they can be unknown.

Assumption 4. and its first derivative are known and bounded, while and its first four derivatives are continuous and bounded.

2.2. Description of RBF NNs

In this paper, RBF NNs will be employed to approximate unknown functions. Mathematically, an RBF NN can be expressed as

where and are the NN outputs and input, is the weight vector, and is the basis function vector with commonly chosen as the Gaussian functions:

where and are constants called the center and width of the basis function, respectively.

Lemma 5 (see [17]). Given any continuous function with a compact set and any constant , by appropriately choosing and , for some sufficiently large integer , there exists an RBF NN such that

where is the optimal weight vector defined as

and denotes the approximation error.

3. Adaptive Neural Controller Design

3.1. Performance and Error Transformation Functions

Let the tracking error be defined as

where is the desired trajectory. Similar to [23, 24], the mathematical expression of the prescribed tracking performance is given by

where and are given positive constants and the smooth function is given by

in which is the initial value of ,   represents the value of at the steady state, and is the decreasing rate of . Then, introduce the following error transformation:

where is the transformed error and is a smooth, strictly increasing, and thus invertible function possessing the following properties:

Note that if is kept bounded, we have , and thus (12) holds. The inverse transformation of can be written as

In this paper, we choose

Differentiating (17) yields

where and . From the properties of the transformation, it is clear that and are bounded and .

Remark 6. From (12) and (13), one can see that and serve as the upper bound of the overshoot and the lower bound of the undershoot of , respectively, the decreasing rate of introduces a lower bound of the convergence rate of , and represents the maximum allowable size of the steady-state value of . Note that , , and should be properly chosen such that .

3.2. Attitude Controller Design via Backstepping

After the error transformation (18), the altitude subsystem (5) is equivalent to

where and . The stabilization of the transformed system (19) is sufficient to guarantee the prescribed tracking performance of system (5).

Based on the backstepping approach, a trajectory tracking controller is designed for the dynamics model given in (19). The design procedure contains 4 steps, and the actual control law will be deduced at the last step. For convenience, let and denote the unknown function to be estimated by RBF NNs and the corresponding compact set in the th step, respectively. Then by using Lemma 5, we have

where and denote the vector valued function and the RBF NN input in the step with proper dimensions that are given below.

Step  1. Let given by (19) be the first error variable. Define , where is the first virtual control signal. Then the derivative of can be expressed as

where and . Since is unknown, we employ an RBF NN to approximate it on a compact set . By properly choosing the basis function vectors we have

where is a positive constant. With respect to the unknown optimal weight vector in (22), define

Besides, let be the estimation of and . Consider the first Lyapunov function

Taking the time derivation of (24) yields

Using Young’s inequality and (23), it can be verified that

Thus, (25) can be rewritten as

which suggests that we choose the first virtual control signal as

Let

where , , and are positive design parameters. Then substituting (28) and (29) into (27), we get

Step i . Define , where is the th virtual control signal. Then the time derivation of is

where and . Consider the th Lyapunov function

where is a positive design parameter, with . By taking the time derivation of (32), we have

Similar to (26), we have

Choose the th virtual control signal

where is updated by

with , , and being positive design parameters. Substituting (35), (36), and (30) into (34), we get

Step  4. The time derivative of is

where and . Let

where is a positive design parameter and with . Differentiating (39) we have

Similar to (26), (40) can be rewritten as

Choose the control signal

where is updated by

with , , and being positive design parameters. Substituting (42), (43), and (37) into (41), we arrive at

Remark 7. The RBF NNs are used to compensate for the complex and uncertain terms to solve the problem of controller complexity, and the repeated derivation of virtual control signal can be avoided. Compared with the neural based control [16, 21], in each design step, by using the estimation of the norm of the NNs weights, only one parameter needs to be updated online; therefore the design procedure can be greatly simplified and the computational burden is greatly reduced. Moreover, the lower bound of the virtual control coefficient is used to avoid the singularity problem without any additional effort.

Remark 8. Since the approximation ability of RBF NNs is on a compact set, we can only guarantee the semiglobal stability of the control scheme.

Theorem 9. Consider system (5) under the Assumptions 2 and 4, with the error transformation (17), the virtual control signals (28) and (35), the control law (42), and the adaptive laws (29), (36), and (43). Then all closed-loop signals are uniformly bounded and the prescribed tracking performance (12) can be guaranteed.

Proof. Using the following facts:

we rewrite (44) as

where

Let

Then we have

Solving (49) gives

which implies that , , , and are bounded. Since is bounded, according to the error transformation of (15), to (17) we can obtain that ; as a result, we have ; that is, the prescribed tracking performance is guaranteed. This completes the proof.

3.3. Velocity Controller Design via Dynamic Inversion

The velocity subsystem of (1) can be rewritten as follows:

where and are unknown nonlinear function . Then define the velocity tracking error as . According to (17) and (18) we obtain

We assume that . The transformed system dynamics of (52) can be rewritten as

where and . Since is an unknown nonlinear function, we use an RBF NN to approximate it:

The control law and the adaptive update law are designed as follows:

where is the estimate of , and , , and are positive design parameters. Consider the Lyapunov function

Differentiating we have

Substituting (55) into (57) then

with inequation

and then

where

Solving (60) gives

It is clear that , , and are bounded. Owning that is bounded, together with the error transformation of (11) into (17), implies that the prescribed tracking performance is guaranteed.

4. Simulation Results

In this section, the numerical simulation results are presented to show the performance of the control scheme. Simulation of the HFV model is conducted for trimmed cruise conditions of 110000 ft and Mach 15. The parameters of simulation model are taken from [16, 21]. The control objective is to track the step change of 100 ft/s in airspeed and 2000 ft in altitude. Linear command filters are used to generate the differentiable commands:

where is Laplace operator, , , and . and are step commands. The inputs of the RBF NNs are , and and the nonlinear functions are approximated with width . The initial values of the adaptive laws are . In addition we choose the design parameters , , , , , , , , , , , , , , and . The parameters of performance functions are given by , , , , , , and . The initial states are chosen as ft/s, ft,  rad,  rad, and  rad/s.

The simulation results are presented in Figures 1−10. The responses to 100-ft/s step-velocity and 2000-ft step-aliunde command in trimmed condition are depicted in Figures 1–3 and Figures 4−6, respectively. From Figures 2 and 5, we see that the tracking errors performance are guaranteed. Figures 7–10 show the simulation results of altitude tracking with square wave trajectory. From the results of simulations, the maximum value of terms is ; it is much smaller than the term whose minimum value is and the maximum value of flight path angle is less than  rad. Therefore, Assumption 1 is reasonable.

Figure 1: Velocity tracking.

Figure 2: The velocity tracking error and the prescribed error bounds.

Figure 3: Throttle setting .

Figure 4: The altitude tracking.

Figure 5: Altitude tracking error and the prescribed error bounds.

Figure 6: Elevator deflection .

Figure 7: Altitude tracking.

Figure 8: Altitude tracking error and the prescribed error bounds.

Figure 9: Elevator deflection .

(a)

(b)

(c)

Figure 10: (a) Path angel ; (b) attack angel ; (c) pitch rate .

5. Conclusion

An adaptive neural control scheme has been proposed for a class of longitudinal dynamics of a generic hypersonic flight vehicle. We have shown that, by using a new constraint variable, the prescribed tracking performance can be achieved. The unknown nonlinear functions associated with each recursive step of backstepping control were approximated by using RBF NNs. For each design step, only one parameter needs to be updated online. Thus the explosion of the complex problem in backstepping control scheme and the computational burden can be greatly reduced. Numerical simulations revealed that the tracking error clearly satisfies the prescribed performance specification and verified the proposed design scheme. Currently, we assume that all of the system states are available and the controller is based on state feedback. However, some states cannot be obtained in some circumstances, especially when the sensor fault occurs. As a result, future work will be focused on output feedback control law design.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61374048), the Research Fund for Doctoral Program of Higher Education of China (Grant no. 20121102110008), and the China Postdoctoral Science Foundation (Grant no. 2013M540839).

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