Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle

Available online at www.sciencedirect.com

Journal of the Franklin Institute 349 (2012) 559–577 www.elsevier.com/locate/jfranklin

Adaptive sliding mode tracking control for a flexible air-breathing hypersonic vehicle$ Xiaoxiang Hua,b, Ligang Wua, Changhua Hub, Huijun Gaoa,n a

Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, PR China b 302 Unit, Xi’an Research Institute of High-tech, Xi’an 710025, PR China Received 16 February 2011; received in revised form 29 June 2011; accepted 8 August 2011 Available online 22 August 2011

Abstract This paper is concerned with the adaptive sliding mode control (ASMC) design problem for a flexible air-breathing hypersonic vehicle (FAHV). This problem is challenging because of the inherent couplings between the propulsion system, the airframe dynamics and the presence of strong flexibility effects. Due to the enormous complexity of the vehicle dynamics, only the longitudinal model is adopted for control design in the present paper. A linearized model is established around a trim point for a nonlinear, dynamically coupled simulation model of the FAHV, then a reference model is designed and a tracking error model is proposed with the aim of the ASMC problem. There exist the parameter uncertainties and external disturbance in the model, which are not necessary to satisfy the so-called matched condition. A robust sliding surface is designed, and then an adaptive sliding mode controller is designed based on the tracking error model. The proposed controller can drive the error dynamics onto the predefined sliding surface in a finite time, and guarantees the property of asymptotical stability without the information of upper bound of uncertainties as well as perturbations. Finally, simulations are given to show the effectiveness of the proposed control methods. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

$ This work was partially supported by National Natural Science Foundation of China (61174126, 90916005, 61025014 & 60736026), Aviation Science Fund of China (2009ZA77001), and the Natural Science Foundation of Heilongjiang Province of China (F201002). n Corresponding author. E-mail address: [email protected] (H. Gao).

0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.08.007

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

560

Nomenclature CD ða, de Þ drag coefficient CDai ith order coefficient of a contribution to CD ða, de Þ di

CDe ith order coefficient of de contribution to CD ða, de Þ constant term in CD ða, de Þ CD0 CL ða, de Þ lift coefficient CLai ith order coefficient of a contribution to CL ða, de Þ de coefficient of de contribution to CL ða, de Þ CL constant term in CL ða, de Þ CL0 CM,Q ða,QÞ contribution to moment due to pitch rate CM, a ðaÞ contribution to moment due to angle of attack CM, de ðde , dc Þ control surface contribution to moment ai CM, ith order coefficient of a contribution to CM, a ðaÞ a 0 CM, constant term in CM, a ðaÞ a CTai ðFÞ ith order coefficient of a in T c mean aerodynamic chord cc Canard coefficient in CM, de ðde , dc Þ ce elevator coefficient in CM, de ðde , dc Þ D drag g acceleration due to gravity h altitude Iyy moment of inertia L left Lv vehicle length M pitching moment m vehicle mass Ni ith generalized force a Ni j jth order contribution of a to Ni Ni0 constant term in Ni N2de contribution of de to N2 Q pitch rate q dynamic pressure S reference area T trust V velocity x state of the control-oriented model a angle of attack bi ðh,qÞ ith thrust fit parameter g flight path angle, g ¼ y  a dc Canard angular deflection de elevator angular deflection x damping ratio for the F dynamics xi damping ratio for elastic mode Zi Zi ith generalized elastic coordinate

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

y li

r F c~ i

o oi 1=hs

561

pitch angle inertial coupling term of ith elastic mode density of air stoichiometrically normalized fuel-to-air ratio constrained beam coupling constant for Zi natural frequency for the F dynamics natural frequency for elastic mode Zi air density decay rate

1. Introduction Air-breathing hypersonic vehicles (AHVs) are crucial because they may represent a more cost efficient way to make access to space routine, or even make the space travel routine and intercontinental travel as easy as intercity travel [1,2]. Although ordinary rocket-based propulsion systems can reach orbital speeds, they are much more expensive to operate in that they must carry oxygen. Being different from ordinary flight vehicles, AHVs adopt scramjets [3], so the AHVs can carry more payload than rocket-powered ones. AHVs use the technology of airframe integrated with scramjet engine configuration [4], which makes the interactions between the elastic airframe, the propulsion system, and the structural dynamics very strong [5]. In addition, the requirements of flight stability and high speed response, the existence of various random interference factors and large uncertainties make it more difficult [6,7]. Flight control design for AHVs is highly challenging, due to the sensitivity changes in fight condition and the difficulty in measuring and estimating the aerodynamic characteristics of the vehicle. Thus, the problem of flight control design is one of the key techniques to the application of AHVs. Because of the dynamics’ enormous complexity, only the longitudinal dynamics models of AHVs have been used for control design. In [8,9], a comprehensive analytical model of hypersonic vehicles was first developed. This model does not has the flexible models, but is still highly nonlinear, multivariable and has strong couplings between the propulsive and aerodynamic effects [10]. Based on this model, several results are available in the literature, see, e.g. [11–13,15,16]. In addition, due to the slender geometries and light structures of this generic vehicles, significant flexible effects cannot be neglected in the control design [19]. A flexible air-breathing hypersonic flight vehicle (FAHV) model, which includes the flexible dynamics, was developed in [17,18]. For this kind of models, control design and simulation have been studied in resent years. The equations of this models become exceedingly complex when flexibility effects are considered, so these models can be used only for simulations or validation purposes [6,7]. In [19], a control-oriented model was derived for the FAHV models using curve fits calculated directly from the forces and moments included in the truth model, then an approximate feedback liberalization example of control design was given to derive a nonlinear controller. In [20], the authors presented two output feedback control design methods for the FAHV models, and adaptive control techniques were also considered in [21]. In [22], dynamic output feedback techniques was used to provide reference robust velocity and altitude tracking control in the presence of model uncertainties and varying flight conditions, and in [23,24], linear

562

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

controllers with input constraints using on-line optimization and anti-windup techniques were also proposed. More recently, a nonlinear robust adaptive control design method was presented in [25], and in [26], the authors considered the modeling of aerothermodynamics effects and gave a Lyapunov-based tracking controller. In practice, it is difficult to measure or estimate the atmospheric properties and aerodynamic characteristics in the flight envelop of FAHV [27]. Therefore, modeling inaccuracies are always in existence, also, various disturbances are presented. This can result some strong adverse effects on the performance of FAHV control systems. Therefore, robust control has been the main technique used for FAHV flight control [28]. On the other hand, sliding mode control (SMC) is well known for its robustness to deal with parametric uncertainties and external disturbances for dynamic systems [39,35,14]. In real system, the upper bound of uncertainties and disturbances are difficult to obtain, and this will reduce the system’s robustness. Adaptive control law could lead to a stable closedloop system and the deviation from the sliding surface is bounded [29]. The adaptive sliding mode control design method for the stability problem have been discussed in [30–32,41], but the result on tracking problem is very limited. Also, few results are given on the unmatched disturbances. As mentioned above, in [16], an adaptive sliding controller was designed for the model developed in [8,9], but unfortunately, this method cannot be used on the model developed in [17,18], since the model is non-minimum phase. Motivated by the above discussions, in this paper, we propose a robust adaptive SMC design method for the longitudinal model of FAHV with system parameter uncertainties and nonlinear perturbations. Firstly, for the longitudinal motion of the FAHV at a special trim condition, a linearized model is formulated for the control design problem. Then, based on the given reference model, a robust sliding surface is developed and the robustness of the designed sliding surface is also discussed. An adaptive law is proposed such that the tracking dynamic is globally stable without the information of disturbances’ upper bound. Finally, an illustrative example is provided to show the effectiveness and advantage of the proposed control design methods. The rest of this paper is organized as follow. Section 2 presents the model of FAHV and the control objectives of this article. Sections 3 and Section 4 give the main results on the robust adaptive SMC design for the longitudinal model of the FAHV. Simulation results are given in Section 5 and we conclude this paper in Section 6. Notation: The notations used throughout the paper are fairly standard. Throughout this paper, the superscript ‘‘T’’ stands for matrix transposition; and Rn denotes the n-dimensional Euclidean space and Rnm denotes the set of all n  m real matrices; I and 0 denote the identity matrix and zero matrix with compatible dimensions. J  J refers to the Euclidean vector norm or spectral matrix norm. Matrices are assumed to be compatible for algebraic operations if their dimensions are not explicitly stated.

2. Model description and control objectives The hypersonic vehicle model considered in this paper is developed by Bolender and Doman [17,18]. Flexibility effects are included in the equations of motion for the longitudinal dynamics of FAHV, by modeling the vehicle as a single flexible structure, whereas the scramjet engine model is adopted from Chavez and Schmidt [9]. A longitudinal sketch of the vehicle is given in Fig. 1. The nonlinear equations is described

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

563

Lv Elevator δe

xB zB M∞

α

Oblique Shock

Bow Shock

Shear Layer

Fig. 1. Geometry of the flexible hypersonic vehicle model.

as follows: 8 > h_ ¼ V sinðy  aÞ, > > > > 1 > > > V_ ¼ ðTcos a  DÞ  g sinðy  aÞ, > > m > > > > 1 1 > > ðT sin a  LÞ þ Q þ cosðy  aÞ, a¼ > > mV V > > > < y_ ¼ Q, ~ Z€ , ~ Z€ þ c > _ ¼M þc Iyy Q > 1 1 2 2 > > > > ~ ~ > > ~ M  c 2 c 1 Z€ 1 , > > k1 Z€ 1 ¼  2B1 o1 Z_ 1  o21 Z1 þ N1  c > 1 > Iyy Iyy > > > > > > c~ 1 c~ 2 Z€ 2 > 2 ~ M > € _ > : k2 Z 2 ¼  2B2 o2 Z 2  o2 Z2 þ N2  c 2 Iyy  Iyy ,

ð1Þ

where k1 ¼ 1 þ

k2 ¼ 1 þ

c~ 1 ¼

Z

c~ 1 Iyy

c~ 2 Iyy

,

,

0

^ f xff ðxÞ dx, m

Lf

c~ 2 ¼

Z 0

La

^ a xfa ðxÞ dx, m

^ f and m ^ a denote the mass densities of the forebody and aftbody, ff and fa stand for the m mode shapes for the forebody and aftbody, respectively. Because the coupling between the pitch and flexible modes is considered in the flexible structure, the equations are more complex than whose using free–free beam to approximate the structure. As

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

564

mentioned in [19], T,L,D,M,N1 and N2 are expressed as L  12rV 2 SC L ða, de Þ, D  12rV 2 SC D ða, de Þ, M  zT T þ 12rV 2 ScðCM, a ðaÞ þ CM, de ðde ÞÞ, 3

2

T  CTa a3 þ CTa a2 þ CTa a þ CT0 , 2

N1  N1a a2 þ N1a a þ N10 , 2

N2  N2a a2 þ N2a a þ N2d de þ N20 , with



 ðh  h0 Þ r ¼ r0 exp , hs CL ¼ CLa a þ CLde de þ CL0 , d2 2

2

CD ¼ CDa a2 þ CDa a þ CDe de þ CDde de þ CD0 , 2

a a 0 CM, a ¼ CM, a a þ CM, a d þ CM, a , 3

CTa ¼ b1 ðh,qÞF þ b2 ðh,qÞ,

CM, de ¼ ce de ,

q ¼ 12rV 2 ,

2

CTa ¼ b3 ðh,qÞF þ b4 ðh,qÞ, CTa ¼ b5 ðh,qÞF þ b6 ðh,qÞ, CT0 ¼ b7 ðh,qÞF þ b8 ðh,qÞ: This model contains nine state variables, that is, ½h,V , a, y,Q, Z1 , Z_ 1 , Z2 , Z_ 2 . The control input F and de do not occur explicitly in the equations of general longitudinal dynamics for the FAHV model in Eq. (1), however, they appear through the forces and moments T, L, D, M, N1 and N2. The nonlinear equation can be expanded by using its Taylor series about the equilibrium point ðx0 ,u0 Þ and keeping only the first order terms: ( _ ¼ AxðtÞ þ BuðtÞ, xðtÞ ð2Þ yðtÞ ¼ CxðtÞ, where xðtÞ ¼ ½h,V , a, y,Q, Z1 , Z_ 1 , Z2, Z_ 2 T is the state of the plant, uðtÞ ¼ ½F, de T is the control input, yðtÞ ¼ ½V ,hT is the output vector. The state x, the input u, and the output y in Eq. (2) are all deviations of the corresponding trajectories of the nonlinear system from a trim condition. Consider the system with parametric uncertainties and disturbances ( _ ¼ ðA þ DAðtÞÞxðtÞ þ BuðtÞ þ f ðxðtÞ,tÞ, xðtÞ ð3Þ yðtÞ ¼ CxðtÞ,

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

565

where DAðtÞ is unknown parameter uncertainties of the matrices A, and f ðxðtÞ,tÞ is an uncertain extraneous disturbance or nonlinearity. For the purpose of model reference, the reference model of the above uncertain system can be defined by x_ m ðtÞ ¼ Am xm ðtÞ þ Bm rðtÞ, n

ð4Þ m

where xm ðtÞ 2 R is the state of the reference model, rðtÞ 2 R is the known, piecewise continuous and bounded reference input to the reference model. Am and Bm are known and real constant matrices with approximate dimensions. In addition, Am is assumed to be stable. The tracking error is defined as eðtÞ ¼ xðtÞ  xm ðtÞ:

ð5Þ

Differentiating Eq. (5) with respect to time and using (3) and (4), we obtain the dynamic equation of tracking error as e_ ðtÞ ¼ Am eðtÞ þ BuðtÞ þ ðA þ DAðtÞ  Am ÞxðtÞ þ f ðxðtÞ,tÞ  Bm rðtÞ, ¼ Am eðtÞ þ BuðtÞ þ ðA  Am ÞxðtÞ  Bm rðtÞ þ DAðtÞxðtÞ þ f ðxðtÞ,tÞ, ¼ Am eðtÞ þ BuðtÞ þ ðA  Am ÞxðtÞ  Bm rðtÞ þ cðxðtÞ,tÞ,

ð6Þ

where cðxðtÞ,tÞ ¼ DAðtÞxðtÞ þ f ðxðtÞ,tÞ. To ensure the achievement of model reference’s objective, the following assumptions are necessary. Assumption 1. The pairs ðAm ,BÞ and ðA,BÞ are controllable. Assumption 2. Matrix B is of full column rank. Assumption 3. There exist matrices M and G such that A  Am ¼ BM, Bm ¼ BG: ð7Þ Assumption 4. DAðtÞ and f ðxðtÞ,tÞ are continuous, and there exist an unknown positive scalar r, such that JcðxðtÞ,tÞJrrJeðtÞJ. Remark 1. The conditions of Assumption 3 are so called matched conditions. Since the reference model is ‘‘free design parameters’’, we can always design a reference model which satisfies Assumption 3. Remark 2. As described in Assumption 4, DAðtÞ and f ðxðtÞ,tÞ are not necessary to satisfy the matched conditions. Then the Eq. (6) can be rewritten as e_ ðtÞ ¼ Am eðtÞ þ BðuðtÞ þ MxðtÞ  GrðtÞÞ þ cðxðtÞ,tÞ:

ð8Þ

The required system performance is to minimize the error between the model states and reference states. Then the main objective of this paper is to design a model reference SMC law such that the states of Eq. (3) track those of the reference model (4) in the asymptotic sense. In addition, the globally asymptotic stability of the system (6) should be guaranteed in the exist of parameter uncertainties and extraneous disturbance. Moreover, since the information of upper bound of cðxðtÞ,tÞ is unknown, an adaptive controller is designed by

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

566

utilizing the Lyapunov stability theorem, and it ensures that the states of Eq. (3) can track those of the reference model. 3. Robust stability analysis of the sliding mode To stabilize the tracking error’s dynamic (8), the SMC technique is utilized here. First, a sliding mode surface function is designed as

sðeðtÞ,tÞ ¼ SeðtÞ ¼ BT P1 eðtÞ,

ð9Þ

nn

where P 2 R is a positive-definite matrix which needs to be designed. Similar to the methods in [38], a transformation matrix is defined as follow: " # " # T ~ 1 B~ T T1 ðB~ PBÞ T¼ ¼ , ð10Þ 1 T2 ðBT P1 BÞ BT P1 where B~ is any basis of the null space of BT. For a given matrix B, B~ is non-unique, but any choice satisfying the condition is acceptable. T1 2 RðnmÞn , T2 2 Rmn , and T 1 ¼ ½PB~ B. Then, through the transformation " # z1 ðtÞ zðtÞ ¼ ¼ TeðtÞ, ð11Þ z2 ðtÞ " eðtÞ ¼ ½PB~ B

z1 ðtÞ

#

z2 ðtÞ

,

ð12Þ

where z1 ðtÞ 2 Rnm and z2 ðtÞ 2 Rm , a ‘‘regular form’’ [37] of the original system can be obtained as " # " #   " # #" z_ 1 ðtÞ T1 T1 Am PB~ T1 Am B z1 ðtÞ 0 ¼ þ uðtÞ þ cðxðtÞ,tÞ: ð13Þ z_ 2 ðtÞ T2 T2 Am PB~ T2 Am B z2 ðtÞ I Corresponded to the regular form, the sliding surface can be rewritten as

sðzðtÞ,tÞ ¼ BT P1 eðtÞ ¼ BT P1 ðT 1 zðtÞÞ ¼ BT P1 Bz2 ðtÞ,

ð14Þ

and the reduced-order sliding mode dynamics on the sliding surface sðzðtÞ,tÞ ¼ 0 with dimension of nm can also be obtained as T

T

T

T

~ 1 B~ Am PBz ~ 1 ðtÞ þ ðB~ PBÞ ~ 1 B~ cðxðtÞ,tÞ: z_ 1 ðtÞ ¼ ðB~ PBÞ

ð15Þ

In the following, a robust stability criterion will be proposed to guarantee the above sliding mode dynamics to be asymptotically stable under the existing of cðxðtÞ,tÞ. Before proceeding, the following lemma will be used to prove our main results. Lemma 1. Let X and Y are real matrices (or vectors) of appropriate dimensions, for any scalar e40, we have X T Y þ Y T X reX T X þ e1 Y T Y :

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

567

Theorem 1. The reduced order sliding mode dynamics in Eq. (15) is asymptotically stable if there exists a symmetric positive-definite matrix P40, and a scalar e40, satisfying " T # T B~ ðAm P þ PATm þ er2 IÞB~ B~ P o0: ð16Þ PB~ eI Proof. Construct a Lyapunov function as T

~ 1 ðtÞ: V ðtÞ ¼ zT1 ðtÞðB~ PBÞz

ð17Þ

By using Lemma 1 and taking the time derivative along the trajectory of the sliding mode dynamics described in Eq. (15), we have T ~ 1 B~ T Am PBÞ ~ T þ cT ðxðtÞ,tÞððB~ T PBÞ ~ 1 B~ T ÞT ðB~ T PBÞz ~ 1 ðtÞ V_ ðtÞ ¼ ½zT1 ðtÞððB~ PBÞ T ~ B~ T PBÞ ~ 1 B~ T Am PBz ~ 1 ðtÞ þ ðB~ T PBÞ ~ 1 B~ T cðxðtÞ,tÞ, þzT1 ðtÞðB~ PBÞ½ð T T ~ 1 ðtÞ þ cT ðxðtÞ,tÞBz ~ 1 ðtÞ þ zT ðtÞB~ T cðxðtÞ,tÞ, ¼ zT ðtÞðB~ PAT B~ þ B~ Am PBÞz 1

m

1

where T

~ 1 ðtÞ þ zT ðtÞB~ cðxðtÞ,tÞrecT ðxðtÞ,tÞcðxðtÞ,tÞ þ e1 ðBz ~ 1 ðtÞÞT ðBz ~ 1 ðtÞÞ, cT ðxðtÞ,tÞBz 1 ~ 1 ðtÞ: ~ 1 ðtÞÞT ðBz ~ 1 ðtÞÞ,rer2 zT ðtÞðPBÞ ~ T ðPBÞz ~ 1 ðtÞ þ e1 zT ðtÞB~ T Bz oer2 eT ðtÞeðtÞ þ e1 ðBz 1 1 Then, we have V_ ðtÞrzT ðtÞXz1 ðtÞ, 1

ð18Þ

where T T ~ T ðPBÞ ~ þ e1 B~ T B~ X ¼ B~ PATm B~ þ B~ Am PB~ þ er2 ðPBÞ T ~ ¼ B~ ðAm P þ PATm þ er2 PP þ e1 IÞB:

ð19Þ

If Xo0, V_ ðtÞr0 is satisfied, which implies that the sliding mode dynamics (15) is asymptotically stable. By Schur complement, Xo0 is equivalent to LMI (16), then the proof is completed. & Further we consider the special case that the uncertain and disturbance satisfy the matched condition, that is, there exist continuous functions H(t) and JðxðtÞ,tÞ of appropriate dimensions such that DAðtÞ ¼ BHðtÞ,

cðxðtÞ,tÞ ¼ BJðxðtÞ,tÞ:

ð20Þ

Then the Eq. (6) can be rewritten as e_ ðtÞ ¼ Am eðtÞ þ BðuðtÞ þ MxðtÞ  GrðtÞ þ HðtÞxðtÞ þ JðxðtÞ,tÞÞ,

ð21Þ

and the LMIs (16) can be reduced to T ~ B~ ðAm P þ PATm ÞBo0:

As described in [38], the above inequality is more easier to be solved.

ð22Þ

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

568

4. Adaptive SMC synthesis In the above section, a sufficient condition to ensure the asymptotic stability of the sliding mode dynamics is discussed. After designing the sliding surface, the next phase of the traditional SMC is to design an appropriate SMC law such that the error dynamics will be driven onto the sliding surface, and remain on it. When utilizing the conventional SMC technique, it is necessary to have the information of the upper bound of cðxðtÞ,tÞ in order to design a control law with switching part dominating the influence of perturbations [33,34]. However, in real system, this bound is difficult to obtain. To overcome this, in this section, an adaptive design method will be proposed. From Assumption 4, we know that, JcðxðtÞ,tÞJrrJeðtÞJ. But the information of r is unknown. Therefore, to obtain the value of r, we should first design an adaptive law to estimate it, thus giving an adaptive sliding mode controller for system (8). Let r^ ðtÞ represent the estimate of r. The corresponding estimation error is:

r~ ðtÞ ¼ r  r^ ðtÞ,

ð23Þ

Theorem 2. Consider the uncertain tracking dynamic systems (8) under Assumptions 1–4, and suppose that the sliding surface is chosen as (9) with P can be solved by LMI (16). Then, the error dynamic can be driven onto the sliding surface (9) and the motion of Eq. (8) remains on the sliding surface with the following adaptive SMC control law: uðtÞ ¼  MxðtÞ þ GrðtÞ  ðBT P1 BÞ1 BT P1 Am eðtÞ ððBT P1 BÞ1 JBT P1 JJeðtÞJr^ ðtÞÞsignðsðeðtÞ,tÞÞ, the adaptive law is designed as r^_ ðtÞ ¼ qJsT ðeðtÞ,tÞJJBT P1 JJeðtÞJ,

ð24Þ

ð25Þ

with r^ ð0Þ ¼ 0, where q40 is a positive constant as the adaptive gain. Proof. Define a Lyapunov function as 1 T 1 ðs ðeðtÞ,tÞsðeðtÞ,tÞ þ r~ 2 ðtÞ: 2 q _ _ Notice that r~ ðtÞ ¼  r^ ðtÞ, and taking the time derivative of V(t), we have V ðtÞ ¼

1 V_ ðtÞ ¼ sT ðeðtÞ,tÞs_ ðeðtÞ,tÞ  r~ ðtÞr^_ ðtÞ q 1 ¼ sT ðeðtÞ,tÞBT P1 ðAm eðtÞ þ BðuðtÞ þ MxðtÞ  GrðtÞÞ þ cðxðtÞ,tÞÞ  r~ ðtÞr^_ ðtÞ q T T 1 T 1 ¼ s ðeðtÞ,tÞðJB P JJeðtÞJr^ ðtÞsignðsðeðtÞ,tÞÞ þ B P cðxðtÞ,tÞÞ r~ ðtÞJsT ðeðtÞ,tÞJJBT P1 JJeðtÞJrJsT ðeðtÞ,tÞJJBT P1 JrJeðtÞJ r~ ðtÞJsT ðeðtÞ,tÞJJBT P1 JJeðtÞJ þ ðr  r^ ðtÞÞJsT ðeðtÞ,tÞJJBT P1 JJeðtÞJ: and V_ ðtÞr0 holds when the control law in Eq. (24) with the adaptive gain in Eq. (25) is applied. Therefore, we can conclude that the adaptive control law (24) can drive the state trajectories of the system in Eq. (8) onto the sliding surface (9). This completes the proof. & Remark 3. Chattering is a common phenomenon in sliding mode control system [33]. To reduce the chattering phenomenon which is usually caused by signðsðeðtÞ,tÞÞ in the sliding

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

569

mode controller of Eq. (24), one simple but useful way utilized here is to replace the term signðsðeðtÞ,tÞÞ by sðeðtÞ,tÞ=ðJsðeðtÞ,tÞJ þ dÞ, where d is an adjustable scalar. By choosing a big value of d, the chattering phenomenon of SMC can be effectively reduced, however, the tracking accuracy will be reduced. To get a better tracking performance and reduce the chattering simultaneously, one can adjust appropriately the parameters q and d.

Table 1 Trim condition. State

Value

h V

85 000 ft 7702:0808 ft s1 1.51531 1.51531 01 s1 1.5122 0 1.2144 0 0.2514 11.46351

a y Q

Z1 Z_ 1 Z2 Z_ 2 F de

9.6 Altitude Change, ft

Velocity Change, ft/s

10000 9500 9000 8500 8000

Velocity change Reference command

7500 0

100

200

x 104

9.4 9.2 9 8.8 8.6

Altitude change Reference command

8.4

300

0

100

t (s)

200

300

200

300

t (s)

Altitude Tracking Error, ft

Velocity Tracking Error, ft/s

x 10−12 8

1 0 −1 −2 −3 −4

6 4 2 0 −2 −4

−5 0

100

200

300

0

100

t (s)

Fig. 2. Case I: tracking performances of closed-loop.

t (s)

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

570

5. Simulation results In this section, an example is provided to illustrate the effectiveness of the robust adaptive sliding mode control proposed in the previous sections. The hypersonic vehicle model parameters are borrowed from [19]. The equilibrium point of the nonlinear vehicle dynamics described by the system in Eq. (1) is listed in Table 1. By using the parameters, the matrices A and B in Eq. (2) can be written as 2

0 6 0:2433  103 6 6 6 1:769  107 6 6 0 6 6 3:964  107 A¼6 6 6 0 6 6 6 0 6 6 0 4 0

"

0 B¼ 0

24:42 37:2

0 0:1349  102 1:027  106 0 2:199  106 0

7702 20:95 0:06961 0 2:946 0

7702 31:92 0 0 0 0

0 0 0

4648 0 2598

0 0 0

8:384  105 0:01122

0 0

0:1226 0 1:491 0

0 0 1 1 0 0

0 0 0 0 0 0

0 0 0 0 0 1

0 272:3 0:66 0 0 0 0 0 0

0 0

0 0

0 1247

0 0 0 0 0 0

0 0 0 0 0 0

0 0 400

7 7 7 7 7 7 7 7 7, 7 7 7 7 0 7 7 7 1 5 0:8

#T :

ð26Þ

Angle of Attack, deg

2.5 2 1.5 1 0.5 0

50

100

150 t (s)

200

250

300

0

50

100

150 t (s)

200

250

300

Pitch Angle, deg

2.5

2

1.5

1

Fig. 3. Case I: angle of attack and flight path angle.

3

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

571

The main control objective is to track a step signal (predefined) with respect to a trim condition, so the reference input r(t) of reference model is chosen as a step input. Each command will pass through a prefilter as HðsÞ ¼

s2

o2n , þ 2zon s þ o2n

where z denotes damping ratio, on stands for natural frequency, and they are assumed to be 0.95 and 0.03 rad/s, respectively. The output of the prefilter is defined as a reference input of the reference model. Based on the method proposed in [36], the matrices Am and Bm of the reference model are chosen as 2 0 0 7702 7702 0 0 6 3:162 0:62 18253:45 21295:5 441:1 90:6 6 6 6 1:434  104 1:633  104 0:881 1:055 0:949 1:165  104 6 6 0 0 0 0 1 0 6 6 0:03336 0:01556 210:91 236:47 8:23 0:477 Am ¼ 6 6 6 0 0 0 0 0 0 6 6 6 0 0 4648 0 0 272:3 6 6 0 0 0 0 0 0 4 0 0 2598 0 0 0 0 6:95

0 1:9

4:311

8:329  104

1 0:0326

0 0:106

0:66 0

0 0

0

400

0:8

" Bm ¼

0 7:5

3

7 7 7 3:75  105 7 7 7 0 7 7, 0:036 7 7 7 0 7 7 5 1

ð27Þ

0

3:165

1:437  104

0

0:0336

0 0

0

16:99

0

0:599

1:65  104

0

0:0158

0 0

0

17:95

#T :

ð28Þ

The modeling of parameter uncertainties is similar to [12], and in this work, the d2

a , Ca , C0 , C parameters of (CLa , CLde , CL0 , CDa , CDa , CDe , CDde , CD0 , CM, M, de ) are M, a a M, a assumed to be uncertain, and these uncertainties are assumed to lie within 710% of nominal values, respectively. The uncertainty of S lies within 75% of nominal value, so does the mean aerodynamic chord c. According to [40], the disturbance f ðxðtÞ,tÞ are assumed to be bounded, which can be regarded as a gust of wind in aerospace. Here we chose 2

2

f ðxðtÞ,tÞ ¼ ½2 sinðtÞ 5 cosðtÞ 0:3 sinðtÞ 0:3 sinðtÞ 0:5 sinðtÞ 0 0 0 0T :

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

572

Then the uncertainty matrix 9DAðtÞ9max can be gotten, where 2

0 6 2:4347  105 6 6 6 1:77  104 6 6 0 6 6 6 9DAðtÞ9max ¼ 6 3:61  107 6 0 6 6 6 0 6 6 0 4 0

0 1:35  104

0 0:0297

0 3:55  104

0 0

0 0

0 0 0 0

4:91  108

0:007

0

0

0

0 0

0

0

0

0

0

0 0

6

2:005  10

0:0025

0

0

0

0 0

0 0

0 0

0 0

0 0

0 0

0 0 0 0

0

0

0

0

0

0 0

0

0

0

0 0

8

0

9  10

3 0 07 7 7 07 7 07 7 7 07 7: 7 07 7 07 7 7 05 0

Chose e ¼ 1, and solve the LMI condition (16) in Theorem 1, we have 2 6 6 6 6 6 6 6 6 P¼6 6 6 6 6 6 6 6 4

0:7

0:03 0:0021

0:0022

0:0043

0:038

n

1:65

0:0005

0:00047

0:0033

0:043

n

n

0:00015

0:000154

0:00035

0:00266

n

n

n

0:00015

n

n

n

n

0:00035 8:23

0:00264 0:477

n

n

n

n

n

0:0057

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

n

0:088

0:073

0:25

0:2034 7 7 7 0:0054 0:0008089 0:00169 7 7 0:0055 0:00076 0:0018 7 7 7 0:0326 0:106 0:036 7 7 7 0:018 0:0021 0:008 7 7 0:982 0:024 0:2157 7 7 7 n 0:065 0:111 5 0:043

0:048

n

n

2:018

Fuel−to−Air Ratio

1.5

1

0.5

0

Elevator Deflection, deg

3

0

50

100

150 t (s)

200

250

300

0

50

100

150 t (s)

200

250

300

15 14 13 12 11

Fig. 4. Case I: the inputs of the plant.

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

573

then by Eq. (9), the sliding surface can be computed as sðeðtÞ,tÞ ¼ SeðtÞ  ¼

82:8

6:23

1902:98 1858:85

524978:6

565700:28

9352:5

740:1

7:7285

16967598:36

18355591:9

462033:26

3257:6

345

174:7

3:498

22167:11 2056:42

 eðtÞ,

and then the SMC law designed in Eq. (24) can be obtained as  uðtÞ ¼ 

0:108

0:047

610:38 718:95

11:03

3:68

0:287

0:0348 0:308



xðtÞ 0:0143 89:31 99:5 4:61 0:017 0:00169 0:073 0:001  0:046 þ rðtÞ 0:0136 0:0144   0:1087 0:0466 238:29 272:5 16 2:359 0:854 1:58 0:374 eðtÞ þ 0:0136 0:0143 85:57 96:47 4:263 0:0086 0:0029 0:2076 0:0055   19404:59 82:006 sðeðtÞ,tÞ  JeðtÞJr^ ðtÞ, 82:006 8:704 JsðeðtÞ,tÞJ þ d 

0:0136 0:1088

and the parameters are set to be ðq, dÞ ¼ ð2  107 ,0:01Þ. In simulation, to illustrate the effectiveness of the proposed controller, we will use the original nonlinear model (not the linear model) to test the performance of the control system. The parameter uncertainties are set to be 10% or 5% of the nominal case, respectively. Here, we consider two cases. Case I: a climbing maneuver at constant

x 104 8600

Altitude Change, ft

Velocity Change, ft/s

9.5 8400 8200 8000 Velocity change

7800

Reference command

0

100

200

9

Altitude change Reference command

8.5 0

300

50

2

8 Altitude Tracking Error, ft

Velocity Tracking Error, ft/s

t (s)

1.5 1 0.5 0 −0.5 0

100

200 t (s)

300

100 150 200 250 300 t (s)

x 10−12

6 4 2 0 −2 0

100

200 t (s)

Fig. 5. Case II: tracking performances of closed-loop.

300

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

574

Angle of Attack, deg

2.5

2

1.5

1 0

50

100

150 t (s)

200

250

300

0

50

100

150

200

250

300

Pitch Angle, deg

2.5

2

1.5 t (s) Fig. 6. Case II: angle of attack and flight path angle.

dynamic pressure and Case II: a climbing maneuver with longitudinal acceleration using separate reference commands for altitude and velocity. In Case I, the vehicle is initially at the trim condition. The reference href(t) is generated to let the vehicle climb 10 000 ft in about 250 s, whereas the velocity reference is computed according to the relation Vref ðtÞ ¼ ½2q expððhref ðtÞ  h0 Þ=hs Þ=r0 1=2 to maintain constant dynamic pressure at q ¼ 2000 psf throughout the maneuver. The effectiveness and tracking performance of the proposed controller are shown in the Figs. 2–4. From the Fig. 2, we can see that the controller provides stable tracking of the reference trajectories. More specifically, the tracking performances for the velocity and altitude are shown in Fig. 2. Figs. 3 and 4 show, respectively, the angle of attack, flight path angle and the inputs of the plant: fuel-to-air ratio F and elevator deflection de . In Case II, the reference commands for altitude and velocity are chosen as 1000 ft/s and 10 000 ft, respectively. The simulation results are shown in Figs. 5–7. The tracking performances for the velocity and altitude are shown in Fig. 5 and others are shown in Figs. 6 and 7. Notice from Figs. 2 and 5 that the tracking error remains remarkably small during the whole maneuver. Hence, the robust SMC method can stabilize the nonlinear system in Eq. (1). 6. Conclusion In this paper, a robust adaptive SMC strategy has been presented for the tracking control problem of the longitudinal dynamics of FAHV model. The linearized model and a

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

575

Fuel−to−Air Ratio

1 0.8 0.6 0.4

Elevator Deflection, deg

0.2 0

50

100

0

50

100

150 t (s)

200

250

300

150

200

250

300

15 14 13 12 11 t (s) Fig. 7. Case II: the inputs of the plant.

reference model have been established, and an error dynamic model has been obtained. Then, an adaptive SMC law has been proposed, which can guarantee the global stability of the closed-loop system without the information of the upper bound of the uncertainties and the external disturbance. Simulation results have validated the effectiveness of the proposed control methods.

References [1] J. Bertin, R. Cumming, Fifty years of hypersonics: where we’ve been, where we’re going, Progress in Aerospace Sciences 39 (2003) 511–536. [2] J. Bertin, J. Periaux, J. Ballmann, Advances in Hypersonics, Birkhauser, Boston, MA, 1992. [3] E. Curran, Scramjet engines: the first forty years, Journal of Propulsion and Power 17 (6) (2001) 1138–1148. [4] W. Engelund, Hyper-X aerodynamics: the X-43A sirframe-integrated acramjet propulsion flight-test experiments, Journal of Spacecraft and Rockets 38 (6) (2001) 801–802. [5] B. Fidan, M. Mirmirani, P. Ioannou, Flight dynamics and control of air-breathing hypersonic vehicles: review and new directions, AIAA Paper, AIAA-2003-7081, Norfolk, VA, USA, December 2003. [6] M. Oppenheimer, M. Bolender, D. Doman, Effects of unsteady and viscous aerodynamics on the dynamics of a flexible air-breathing hypersonic vehicle, AIAA Paper, 2007, pp. 2007–6397. [7] T. Williams, M. Bolender, D. Doman, O. Morataya, An aerothermal flexible mode analysis of a hypersonic vehicle, AIAA Paper, 2006, pp. 2006–6647. [8] D. Schmidt, Dynamics and control of hypersonic aeropropulsive/aeroelastic vehicles, AIAA Paper, 1992, pp. 1992–4326.

576

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

[9] F. Chavez, D. Schmidt, Analytical aeropropulsive/aeroelastic hypersonic-vehicle model with dynamic analysis, Journal of Guidance, Control, and Dynamics 17 (6) (1994) 1308–1319. [10] F. Chavez, D. Schmidt, Uncertainty modeling for multivariable-control robustness analysis of elastic highspeed vehicles, Journal of Guidance, Control, and Dynamics 22 (1) (1999) 87–95. [11] D. Schmidt, Optimum mission performance and multivariable flight guidance for airbreathing launch vehicles, Journal of Guidance, Control, and Dynamics 20 (6) (1997) 1157–1164. [12] H. Buschek, A. Calise, Uncertainty modeling and fixed-order controller design for a hypersonic vehicle model, Journal of Guidance, Control, and Dynamics 20 (1) (1997) 42–48. [13] E. Mooij, Numerical investigation of model reference adaptive control for hypersonic aircraft, Journal of Guidance, Control, and Dynamics 24 (2) (2001) 315–323. [14] L. Wu, C. Wang, Q. Zeng, Observer-based sliding mode control for a class of uncertain nonlinear neutral delay systems, Journal of the Franklin Institute 345 (3) (2008) 233–253. [15] R. Lind, Linear parameter-varying modeling and control of structural dynamics with aerothermoelastic effects, Journal of Guidance, Control, and Dynamics 25 (4) (2002) 733–739. [16] H. Xu, M. Mirmirani, P. Ioannou, Adaptive sliding mode control design for a hypersonic flight vehicle, Journal of Guidance, Control, and Dynamics 27 (5) (2004) 829–838. [17] M. Bolender, D. Doman, A non-linear model for the longitudinal dynamics of a hypersonic air-breathing vehicle, in: Proceeding of the 2005 Guidance, Navigation and Control Conference, AIAA Paper, no. 20056255, San Francisco, California, August 2005. [18] M. Bolender, D. Doman, A nonlinear longitudinal dynamical model of an air-breathing hypersonic vehicle, Journal of Spacecraft and Rockets 44 (2) (2007) 374–387. [19] J. Parker, A. Serrani, S. Yurkovich, M. Bolender, D. Doman, Control-oriented modeling of an air-breathing hypersonic vehicle, Journal of Guidance, Control, and Dynamics 30 (3) (2007) 856–869. [20] P. Jankovsky, D. Sigthorsson, A. Serrani, S. Yurkovich, M. Bolender, D. Doman, Output feedback control and sensor placement for a hypersonic vehicle model, in: AIAA Guidance, Navigation and Control Conference and Exhibit, no. 2007–6327, 2007. [21] M. Kuipers, M. Mirmirani, P. Ioannou, Y. Huo, Adaptive control of an aeroelastic airbreathing hypersonic cruise vehicle, AIAA Paper, 2007, pp. 2007–6326. [22] D. Sigthorsson, P. Jankovsky, A. Serrani, S. Yurkovich, M. Bolender, D. Doman, Robust linear output feedback control of an airbreathing hypersonic vehicle, Journal of Guidance, Control, and Dynamics 31 (4) (2008) 1052–1066. [23] K. Groves, A. Serrani, S. Yurkovich, M. Bolender, D. Doman, Anti-windup control for an air-breathing hypersonic vehicle model, AIAA Paper, 2006, pp. 2006–6557. [24] D. Sigthorsson, A. Serrani, S. Yurkovich, M. Bolender, D. Doman, Tracking control for an overactuated hypersonic air-breathing vehicle with steady state constraints, AIAA Paper, 2006, pp. 2006–6558. [25] L. Fiorentini, A. Serrani, M. Bolender, D. Doman, Nonlinear robust adaptive control of flexible airbreathing hypersonic vehicles, Journal of Guidance, Control, and Dynamics 32 (2) (2009) 401–416. [26] Z. Wilcox, W. MacKunis, S. Bhat, R. Lind, W. Dixon, Lyapunov-based exponential tracking control of a hypersonic aircraft with aerothermoelastic effects, Journal of Guidance, Control, and Dynamics 33 (4) (2010) 1213–1224. [27] D. Schmidt, Integrated control of hypersonic vehicles, AIAA Paper, 1993, pp. 93–5091. [28] K. Austin and B. Mechanical, Evolutionary design of robust flight control for a hypersonic aircraft, Ph.D. Thesis, Department of Mechanical Engineering, Queensland University, Australia, 2002. [29] H. Sira, E. Colina, A sliding mode strategy for adaptive learning in adalines, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 42 (12) (1995) 1001–1012. [30] H. Yang, Y. Xia, M. Fu, P. Shi, Robust adaptive sliding mode control for uncertain delta operator systems, International Journal of Adaptive Control and Signal Processing 24 (8) (2010) 623–632. [31] T. Wu, Design of adaptive variable structure controllers for T–S fuzzy time-delay systems, International Journal of Adaptive Control and Signal Processing 24 (2) (2010) 106–116. [32] C. Cheng, S. Chien, Adaptive sliding mode controller design based on T–S fuzzy system models, Automatica 42 (6) (2006) 1005–1010. [33] J. Hung, W. Gao, J. Hung, Variable structure control: a survey, IEEE Transactions on Industrial Electronics 40 (1) (1993) 2–22. [34] K. Yong, V. Utkin, U. Ozguner, A control engineer’s guide to sliding mode control, IEEE Transactions on Control Systems Technology 7 (3) (1999) 328–342. [35] C. Cheng, I. Liu, Design of MIMO integral variable structure controllers, Journal of the Franklin Institute 36 (1999) 1119–1134.

X. Hu et al. / Journal of the Franklin Institute 349 (2012) 559–577

577

[36] C. Dong, Y. Hou, Y. Zhang, Q. Wang, Model reference adaptive switching control of a linearized hypersonic flight vehicle model with actuator saturation, Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering 224 (3) (2010) 289–303. [37] C. Edwards, S. Spurgeon, Sliding Mode Control: Theory and Applications, Taylor & Francis, London, 1998. [38] H. Choi, Variable structure control of dynamical systems with mismatched norm-bounded uncertainties: an LMI approach, International Journal of Control 74 (13) (2001) 1324–1334. [39] M. Chana, C. Taob, T. Lee, Sliding mode controller for linear systems with mismatched time-varying uncertainties, Journal of the Franklin Institute 337 (2–3) (2000) 105–115. [40] T. Gibson, L. Crespo, A. Annaswamy, Adaptive control of hypersonic vehicles in the presence of modeling uncertainties, in: Proceeding of the 2009 American Control Conference, Saint Louis, Missouri, June, 2009, pp. 3178–3183. [41] Y. Xia, Z. Zhu, C. Li, H. Yang, Q. Zhu, Robust adaptive sliding mode control for uncertain discrete-time systems with time delay, Journal of the Franklin Institute 347 (1) (2010) 339–357.