Guaranteeing preselected tracking quality for air-breathing hypersonic non-affine models with an unknown control direction via concise neural control

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Abstract

A simplified neural controller is addressed for the longitudinal dynamics of an air-breathing hypersonic vehicle (AHV) with a completely unknown control direction by utilizing the prescribed performance control scheme. Unlike the existing literatures, the exploited methodology does not require an affine AHV model or any prior information about the sign of control gains. Moreover, the proposed strategy can provide preselected bounds on the transient and steady performance of velocity and altitude tracking errors. The altitude dynamics is converted into a pure feedback formulation with an unknown control direction, based on which, a novel adaptive neural controller that is quite simpler than the ones derived from back-stepping designs is achieved. For the problem of the unknown control direction, a Nussbaum-type function is introduced to handle it. By employing the minimal-learning parameter (MLP) technique to regulate the norm instead of the elements of the ideal weight vector, only one learning parameter is required for neural approximation. Thus, a low computational burden design is obtained. Finally, simulations are performed to verify the presented control approach.

Introduction

Air-breathing hypersonic vehicles (AHVs) are crucial since they can provide a promising technology for cost-efficient access to near-space owing to the fact that the scramjet propulsion may offer significant advantages over the traditional expendable rockets [1], [2]. The flight control design for AHVs is a challenging task because of the unique characteristics of the vehicle dynamics. Due to the underslung location of the scramjet engine, there exist strong couplings between the propulsive and aerodynamic forces [3]. Moreover, the slender geometry and the flexibility of the vehicle structure lead to notable vibrational modes, which may further significantly affect the aerodynamic forces [4], [5]. Besides, significant uncertainties result from the variability of the vehicle characteristics with varying flight conditions, fuel consumption, and thermal effects [6].

Recently, lots of encouraging results on controlling the longitudinal dynamics of AHVs have been reported. In [7], [8], a guaranteed cost control methodology is studied for an AHV with the flexible effects. To handle the problem of parametric uncertainties, sliding mode control (SMC) schemes are addressed for AHVs [9], [10]. Furthermore, a high order SMC strategy [11] with continuous control inputs is presented for an AHV to provide robust tracking of velocity and altitude commands, while the undesired chattering phenomenon is avoided. For the longitudinal dynamics of an AHV subject to actuator faults, several fault tolerant control approaches are proposed [12], [13], [14], [15], [16]. In [17], a combined model predictive control with fault tolerant control methodology is investigated for an AHV. By using additional linear matrix inequalities (LMIs) to address the vehicle input constraints, a robust tracking controller is explored for an AHV with input constraints [18], [19]. Simulation results indicate that excellent tracking performance with good robustness can be achieved by that strategy. In [20], a fuzzy model is firstly constructed via a T–S fuzzy modeling technique and then a robust H dynamic output feedback controller is addressed for the nonlinear longitudinal model of an AHV. For the cruise control of an AHV, a new simplified methodology is proposed by solving a system of linear algebraic equations [21]. In [22], a trajectory linearization control (TLC) approach is presented for the reentry control of an AHV. The robustness of that scheme is guaranteed via SMC and an extended disturbance observer. Different from [22], a novel TLC strategy is exploited via active disturbance rejection control (ADRC) and moreover the problem of actuator saturation is taken into consideration [23].

Back-stepping is evolved as a powerful tool for controlling nonlinear systems with matched or unmatched uncertainties. Also, it has been widely used for AHV control system designs [24], [25], [26], [27], [28]. In [3], it is assumed that all of the coefficients of the AHV model are subject to uncertainties and are estimated via a parameter projection scheme, which ensures the robustness and nonsingularity of the developed control laws. Unfortunately, that control approach may result in quite high computational costs owing to its too many adaptive parameters. Meanwhile, the problem of “explosion of terms” is ignored in that study. Moreover, the Kriging system is used to estimate system uncertainties, while a robust adaptive back-stepping controller is developed for an AHV [24]. More specially, a new nonlinear disturbance observer (NDO) [29] is explored based on the results obtained in [30] to estimate both the external disturbances and parametric uncertainties, on the basis of which, a robust back-stepping control strategy is exploited for an AHV to provide robust tracking of velocity and altitude reference trajectories. In [29], the problem of “explosion of terms” encountered in the traditional back-stepping control is successfully eliminated via a tracking differentiator. It is worth mentioning that improved control with robust performance can be achievable if the unknown nonlinearities are approximated by neural networks (NNs) [31], [32], [33]. In [33], it is assumed that the nonlinear function f(•) and the control gain g(•) are completely unknown and are approached by two different NNs, while an adaptive integral dynamic surface control strategy is proposed for an AHV using back-stepping. Notice the fact that neural approximation may result in computational burdens owing to the large amount of NNs and learning parameters. Thereby, many efforts have been made to reduce the needed NNs and learning parameters. Different from [33], the NN is applied to estimate the explored back-stepping controller rather than the unknown nonlinearities f(•) and g(•) of an AHV model [34]. In this way, for each subsystem, only one NN is required. For an AHV model with dead-zone input nonlinearities and an unknown control direction, a robust adaptive neural back-stepping control scheme is investigated [35]. In that study, only one NN is used to approach the lumped uncertainty of each subsystem based on a strict assumption that g(•) is strictly positive and bounded. In [36], the minimal-learning parameter (MLP) technique is introduced to estimate the norm instead of the elements of the ideal weight vector. As a consequence, the learning parameters are decreased greatly. Besides, the altitude dynamics of an AHV is rewritten a pure feedback affine formulation rather than a strict feedback one [37], [38]. Then, a novel adaptive neural control method is achieved without back-stepping [37], [38].

Despite the previous progress in the flight control of AHVs, further studies are still urgently needed. It is noteworthy that all the above control methodologies are derived from affine models. However, the AHV model has no affine appearance of the control inputs [39], [40]. In this paper, the objective is to exploit a concise prescribed performance neural controller for the longitudinal dynamics of an AHV without using affine models and back-stepping. Firstly, the altitude dynamics is rewritten as a normal non-affine formulation and is further transformed into a pure feedback non-affine one. Then, a novel adaptive neural controller with prescribed performance is constructed based on the Mean Value Theorem. Thirdly, a Nussbaum-type function is introduced to deal with the problem of unknown control directions. For the velocity dynamics, a PI controller is employed. Finally, simulation results illustrate the effectiveness of the proposed control scheme. The special contributions of this paper include the below mentioned points.

  • (1)

    The altitude dynamics is expressed as a pure feedback formulation, based on which, a simplified neural control methodology is achieved such that the complex design process of back-stepping is completely avoided.

  • (2)

    In comparison with the control schemes presented in [31], [33], [34], [35], the proposed controller possesses quite low computational burdens and satisfactory prescribed performance. Throughout this paper, only one NN is applied to approximate the lumped system uncertainty and moreover only one learning parameter is required for neural approximation.

The rest of this paper is structured as follows. In Section 2, the AHV model and preliminaries are briefly introduced. The controller is explored in Section 3. In Section 4, the simulation studies are made. Finally, the conclusions are presented in Section 5.

Section snippets

Model description

The longitudinal dynamic model considered in this paper is developed by Bolender and Doman [39]. The equations of motion are formulated as follows [40]:V̇=Tcos(θγ)Dmgsinγḣ=Vsinγγ̇=L+Tsin(θγ)mVgVcosγθ̇=QQ̇=M+ψ˜1η¨1+ψ˜2η¨2Iyyk1η¨1=2ζ1ω1η̇1ω12η1+N1ψ˜1MIyyψ˜1ψ˜2η¨2Iyyk2η¨2=2ζ2ω2η̇2ω22η2+N2ψ˜2MIyyψ˜2ψ˜1η¨1Iyy

The approximations of T, D, L, M, N1 and N2 are defined as [40]TCTα3α3+CTα2α2+CTαα+CT0Dq¯S(CDα2α2+CDαα+CDδe2δe2+CDδeδe+CD0)Lq¯S(CLαα+CLδeδe+CL0)MzTT+q¯Sc¯[CM,αα2α2+CM,ααα+CM,α0+

Controller design

To facilitate the subsequent control designs, we decompose the vehicle dynamics into the velocity subsystem (i.e., Eq. (1)) and the altitude subsystem (i.e., Eqs. (2), (3), (4), (5)), respectively [25], [26], [27], [28], [29].

Simulation results

The proposed controller is tested in simulation adopting the nonlinear model (1), (2), (3), (4), (5), (6), (7) implemented in MATLAB/SIMULINK®. In this simulation study, the vehicle is assumed to climb a step maneuver from the initial trim conditions, provided in Table 1, to the final values h=85,800 ft and V=8200 ft/s.

The reference trajectories of velocity and altitude are smoothed via the following filter:0.12s2+2×0.9×0.1×s+0.12

The center of X1 is evenly spaced in [7700 ft/s, 8200 ft/s]×[−1 deg, 1 

Conclusions

In this paper, a novel adaptive neural non-affine controller is proposed for the longitudinal dynamics of an AHV with an unknown control direction via prescribed performance control. Compared with the traditional back-stepping control, it is proved that the developed controller exhibits a more simplified control structure and can provide better prescribed performance of altitude and velocity tracking errors. Moreover, the practicability of the design is good and the computational load of the

Acknowledgments

The authors would like to express their sincere thanks to anonymous reviewers for their helpful suggestions for improving the technical note. This work was supported by the National Natural Science Foundation of China (Grant nos. 61573374 and 61503408).

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