Adaptive backstepping control for air-breathing hypersonic vehicle subject to mismatched uncertainties

https://doi.org/10.1016/j.ast.2020.106244Get rights and content

Abstract

This paper studies the tracking control problem of the air-breathing hypersonic vehicle (AHV) subject to mismatched uncertainties. A novel model transformation is proposed for AHV with uncertain parameters, to address the unavailable states problem of conventional model transformation. Then, a novel adaptive backstepping control method is proposed for a class of nonlinear system with mismatched uncertainties, and a projection operator is designed to avoid the singular problem. The adaptive controller guarantees asymptotical command tracking, instead of uniformly ultimate boundness (UUB), for the closed-loop uncertain system. Then, the proposed control scheme is directly applied to the design of flight controller based on the transformed model. Lastly, a detailed simulation is conducted to demonstrate the efficiency and superiority of the proposed method.

Introduction

Air-breathing hypersonic vehicle (AHV) generally refers to the aircraft powered by a scramjet and flying at a velocity greater than Mach 5 [1]. It provides a promising potential for reliable and cost-efficient access to space travel routines, and therefore, attracts great interest in global world [2], [3], [4]. Although having significant advantages, AHV also brings challenges for flight control system design, due to the characteristics of highly nonlinear dynamics, strong couplings and complex uncertainties. Flight control system design is one of key techniques to ensure safety and maneuverability of aircrafts [5], [6]. Therefore, high-performance controller is very important for AHV, and it needs to have enough robustness and adaptive ability to accommodate these characteristics [4].

The linearized model based control methods have been already applied to AHV, such as H design and μ-synthesis [7], composite robust linear output feedback control [8], linear quadratic regulator [9], and other multivariable linear control methods. In presence of the high nonlinearity, there are considerable modeling errors between the linearized model and the original nonlinear model. Consequently, the control performance of linear methods may be severely degraded [2], [10]. To further improve the control performance, nonlinear control methods were introduced to AHV model, such as feedback linearization control [11], sliding mode control [12], [13], neural network control [14] and backstepping control [3], [5], [15], [16].

Backstepping is one of the most widely used nonlinear control methods. It provides systematic procedures for controller design of strict feedback systems. Since the backstepping concept was initially introduced in [17], [18], it has been widely studied and applied to many kinds of control plants such as robot [19], invert pendulum [20], quadrotor [21], AHV [5], [16] and many others.

When backstepping is applied to AHV, for the convenience of controller design, the aircraft model is often divided into velocity subsystem and altitude subsystem [22], [23]. Since the AHV model is a high-order strict feedback system, the “explosion of terms” may happen due to the calculation of high-order differentiation of virtual control laws [24], [1]. Therefore, the dynamic surface [5], [14], [25], [26] and different differentiators [15], [27] are introduced to address this issue. To improve the robustness of backstepping controller to uncertainties in AHV model, many disturbance rejection methods are employed. When disturbance observers are utilized to improve the robustness of controllers, the “separation principle” is still satisfied [28]. Therefore, it is convenient to combine backstepping control with disturbance observers. The conventional and the 2-order nonlinear disturbance observers were employed to estimate the uncertainties in the steps of the backstepping procedure in [24] and [26]; to further improve the convergence speed of estimation errors, ref. [23] designed the fixed-time disturbance observer in backstepping scheme. Neural networks [15], [14], [16], [29] and fuzzy logic systems [30], [31] are also utilized to improve the robustness of backstepping scheme, because of their good global approximation ability for nonlinear functions. Backstepping is also improved to deal with the control problems caused by the actuator of AHV. For examples, the Nussbaum gain technique was introduced to backstepping scheme to handle the unknown control direction in [26]; an auxiliary system was designed to deal with input saturation in [22]; the fault tolerant control methods based on backstepping were studied for actuator faults in [5], [15], [24], [32]. Besides, backstepping scheme is also further improved to solve some specific control problems. The disturbance effect indicator was defined to improve the transient performance in [3]; the fixed-time sliding mode method was combined with backstepping to improve convergence speed in [23]; the novel compensator was proposed to limit the attack angle within the predefined region in [33]; the Lipschitz continuous dead zone modification was designed to enhance the robustness of adaptive backstepping control in [34], and the prescribed performance techniques were designed to keep tracking errors within preset ranges in [15], [22], [24], [35].

How to effectively deal with uncertainties has always been a key issue of flight controller design for AHV. Generally, the uncertainties in a control plant can be classified into two categories: the matched uncertainty and the mismatched uncertainty. The mismatched uncertainty is difficult to deal with, because it doesn't satisfy “matching condition” and thus cannot be compensated directly [36]. Ref. [5] designed adaptive laws to compensate parameter uncertainties of AHV model, and guaranteed uniformly ultimate boundedness (UUB) of tracking errors. Refs. [16] and [35] compensated the mismatched uncertainties of AHV model by adaptively estimating the upper bounds of uncertainties. Refs. [37], [38] eliminated the mismatched uncertainties by estimate the derivatives of system outputs with differentiators. However, the high-order differentiators will usually cause undesirable amplification of noises. Refs. [1], [3], [26], [23], [11], [13] estimated the mismatched uncertainties with disturbance observers. Besides, the mismatched uncertainties are also approximated by radial basis function neural networks (RBFNNs) [15], [14]. However, the estimation errors of disturbance observers and the approximation errors of neural networks are usually inevitable, and therefore asymptotical tracking of command signals is usually not guaranteed.

This paper considers the tracking control problem of AHV model subject to mismatched uncertainties. Compared with the existing literature, the contributions of this paper are summarized as follows.

1) A practical model transformation is proposed for AHV in presence of parameter uncertainties. The conventional model transformation in [39], [40], [41] is initially analyzed, and we find it will generate unavailable states in transformed model when there are uncertain parameters. Therefore, it is not suitable for practical controller design in uncertain parameters case. To overcome this drawback, a novel coordinate transformation is proposed, in which all new states are based on measurable flight states and nominal parameters. Besides, the transformed model has the strict feedback form, and can accommodate system uncertainties well for controller design.

2) A novel adaptive backstepping control method is proposed for a class of high-order nonlinear system with mismatched uncertainties. Mismatched uncertainties can usually be addressed by the combination of backstepping and some other methods, such as disturbance observer [3], [26], [42], neural network [14], [43], [44] and fuzzy logic system [30]. However, only the UUB property is guaranteed for the closed-loop system. Although there are some methods which can make tracking errors converge to zero in mismatched uncertainty cases [36], [45], [31], [13], [11], the control laws are discontinuous [36], [45], [31] or non-smooth [13], and the disturbance needs to assumed to be a constant in the final stage [11]. The proposed control scheme in this paper can utilize smooth control inputs to guarantee asymptotical tracking of command signals. Compared with the previous adaptive backstepping control methods [17], [18], [45], the proposed adaptive backstepping control can deal with the more complex uncertainties; the uncertain parameters of different subsystems needn't to be assumed to be the same, and the uncertainties of control gain matrices are also considered in this paper. According to the design results of the proposed controller, to completely compensate the mismatched uncertainties in a specific subsystem, the adaptive laws need the information of not only states in the subsystem, but also all the states in following subsystems behind it, which reflects the difficulty to eliminate the mismatched uncertainties.

3) A more reasonable flight controller is designed for AHV model subject to complex parameter uncertainties. Based on the proposed model transformation and the proposed adaptive backstepping control method, the flight controller is synthesized. Therefore, the practicality of the flight controller is efficiently improved, since the problem of unavailable states is solved and the tracking errors of command signals are eliminated totally.

Notations

  • It is defined that k=ij()=0, if i>j.

  • yx=[y1x1y1x2y1xny2x1y2x2y2xnymx1ymx2ymxn]x(i)=xi where x=[x1,x2,,xn]TRn and y=[y1,y2,, ym]TRm.

  • The notation xij has the same meaning with xi,j; we will use the former notation for simplicity, and use the latter one when ambiguity may appear.

Section snippets

Hypersonic vehicle model

The hypersonic vehicle model considered in this paper is developed by NASA Langley Research Center [40]. Based on the assumption of horizontal ground, the longitudinal dynamic model of AHV in cruise phase is denoted by the following differential equations{h˙=VsinγV˙=TcosαDmgsinγγ˙=L+TsinαmVgcosγVα˙=qγ˙q˙=MyyIyy where h, V, γ, α and q are flight altitude, flight velocity, flight path angle, attack angle and pitch angle rate, respectively; m,g and Iyy are mass, gravity acceleration and

Nonlinear coordinate transformation

Defining the state vector x=[h,V,γ,α,q,β,β˙]T, the control input vector u=[δe,βc]T, the output vector y=[h,V,]T, the parameter vector p=[m,S,c¯,Iyy, ce,CLα,CDα2, CDα,CD0,CMα,α2, CMα,α,CMα,0,CMq,α2, CMq,α,CMq,0,CTβ,CT0]T and the nominal parameter vector p0=[m0,S0,c¯0,Iyy0, ce0,CL0α, CD0α2,CD0α,CD00,CM0α,α2,CM0α,α, CM0α,0,CM0q,α2,CM0q,α, CM0q,0,CT0β, CT00]T, we can rewrite the nonlinear model of hypersonic vehicle as{x˙=f(x,p)+g(x,p)uy=[V,h]T where g(x,p)=[g1(x,p),g2(x,p)]. The expressions of

A novel adaptive backstepping control method for high-order nonlinear system with mismatched uncertainties

This subsection presents a novel adaptive backstepping control for a class of high-order nonlinear system with both matched uncertainties and mismatched uncertainties. The proposed control method can be directly applied to flight controller design of AHV based on the transformed model (9).

A generic nonlinear system is considered here, given by{x˙1=f1(x¯1)+(g1(x¯1)+δ1)x2+ϕ1T(x¯1)θ1x˙2=f2(x¯2)+(g2(x¯2)+δ2)x3+ϕ2T(x¯2)θ2x˙i=fi(x¯i)+(gi(x¯i)+δi)xi+1+ϕiT(x¯i)θix˙n=fn(x¯n)+(gn(x¯n)+δn)u+ϕnT(x¯n)θny=x

Simulation and discussion

In this section, a detailed numerical simulation is conducted to analyze the effectiveness of the proposed control method on SIMULINK platform of MATLAB. The detailed model of AHV is adopted from [40]. In the simulation, the AHV are flying in cruise phase, and the initial flight states are set as h(0)=33528m, V(0)=4590.3m/s, γ(0)=0rad, α(0)=0rad and q(0)=0rad/s. To demonstrate the robustness of the proposed adaptive backstepping control, 20% parameter uncertainties are considered, which means |Δ

Conclusion

In this paper, the mismatched uncertainties are considered in model transformation and controller design for AHV. In order to obtain a practical control-oriented model accommodating parameter uncertainties, a novel coordinate transformation is introduced to original AHV model. The transformed model avoids the problem of unavailable states in conventional model transformation. A novel adaptive backstepping control scheme is proposed for a class of nonlinear system with mismatched uncertainties.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The research presented in this document is supported by the National Natural Science Foundation of China (Grant numbers 61673209, 61973158, 71971115).

References (45)

  • Y. Ding et al.

    Robust fixed-time sliding mode controller for flexible air-breathing hypersonic vehicle

    ISA Trans.

    (2019)
  • X. Yin et al.

    Disturbance observer-based gain adaptation high-order sliding mode control of hypersonic vehicles

    Aerosp. Sci. Technol.

    (2019)
  • X. Bu et al.

    Tracking differentiator design for the robust backstepping control of a flexible air-breathing hypersonic vehicle

    J. Franklin Inst.

    (2015)
  • X. Bu et al.

    Tracking control of air-breathing hypersonic vehicles with non-affine dynamics via improved neural back-stepping design

    ISA Trans.

    (2018)
  • Y. Zhang et al.

    Exponential sliding mode tracking control via back-stepping approach for a hypersonic vehicle with mismatched uncertainty

    J. Franklin Inst.

    (2016)
  • F. Wang et al.

    Disturbance observer based robust backstepping control design of flexible air-breathing hypersonic vehicle

    IET Control Theory Appl.

    (2019)
  • Y. Wang et al.

    Reliable fuzzy tracking control of near-space hypersonic vehicle using aperiodic measurement information

    IEEE Trans. Ind. Electron.

    (2019)
  • Z. Guo et al.

    Robust tracking for hypersonic reentry vehicles via disturbance estimation-triggered control

    IEEE Trans. Aerosp. Electron. Syst.

    (2020)
  • A.A. Rodriguez et al.

    Modeling and Control of Scramjet-Powered Hypersonic Vehicles: Challenges, Trends, Tradeoffs

    (2008)
  • Q. Hu et al.

    Adaptive control for hypersonic vehicles with time-varying faults

    IEEE Trans. Aerosp. Electron. Syst.

    (2018)
  • H. Buschek et al.

    Uncertainty modeling and fixed-order controller design for a hypersonic vehicle model

    J. Guid. Control Dyn.

    (1997)
  • D.O. Sigthorsson et al.

    Robust linear output feedback control of an airbreathing hypersonic vehicle

    J. Guid. Control Dyn.

    (2008)
  • Cited by (26)

    • Improved adaptive backstepping control of MPCVD reactor systems with non-parametric uncertainties

      2023, Journal of the Franklin Institute
      Citation Excerpt :

      As a mature design tool in model based nonlinear control, the adaptive backstepping technique has been applied to a wide range of nonlinear systems for decades. For instance, in [5], the authors solved the tracking control problem of the air-breathing hypersonic vehicle subject to mismatched uncertainties by using a novel adaptive backstepping control method. An adaptive backstepping controller with uncertainty and disturbance estimation was proposed in [6] for a generic horizontal marine turbine, and the high-efficiency control for maximizing the marine current power generation was addressed.

    • Incremental sliding-mode control and allocation for morphing-wing aircraft fast manoeuvring

      2022, Aerospace Science and Technology
      Citation Excerpt :

      By regarding morphing as an auxiliary control and considering model uncertainties due to fast morphing, the robust control problem of morphing aircraft is a control problem for an overactuated system with model uncertainties and disturbances [22,23]. In addition, the model uncertainties contain matched and mismatched uncertainties [24]. To attenuate model uncertainties and alleviate the chattering phenomenon in practical applications, a nonlinear disturbance-observer-based sliding-mode control (NDOSMC) [25] were developed based on a system model.

    View all citing articles on Scopus
    View full text