Continuous high order sliding mode controller design for a flexible air-breathing hypersonic vehicle

ISA Transactions 53 (2014) 690–698

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Research Article

Continuous high order sliding mode controller design for a flexible air-breathing hypersonic vehicle Jie Wang a,n, Qun Zong b, Rui Su b, Bailing Tian b a b

School of Control Science and Engineering, Hebei University of Technology, Tianjin 300130, China School of Electrical and Automation Engineering, Tianjin University, Tianjin 300072, China

art ic l e i nf o

a b s t r a c t

Article history: Received 31 January 2013 Received in revised form 22 December 2013 Accepted 5 January 2014 Available online 16 February 2014 This paper was recommended for publication by Jeff Pieper.

This paper investigates the problem of tracking control with uncertainties for a flexible air-breathing hypersonic vehicle (FAHV). In order to overcome the analytical intractability of this model, an Input– Output linearization model is constructed for the purpose of feedback control design. Then, the continuous finite time convergence high order sliding mode controller is designed for the Input–Output linearization model without uncertainties. In addition, a nonlinear disturbance observer is applied to estimate the uncertainties in order to compensate the controller and disturbance suppression, where disturbance observer and controller synthesis design is obtained. Finally, the synthesis of controller and disturbance observer is used to achieve the tracking for the velocity and altitude of the FAHV and simulations are presented to illustrate the effectiveness of the control strategies. & 2014 ISA. Published by Elsevier Ltd. All rights reserved.

Keywords: Flexible air-breathing hypersonic vehicle Continuous high order sliding mode control Sliding mode disturbance observer Finite-time stability

1. Introduction Air-breathing hypersonic vehicles (AHVs) are crucial to the development of new technologies in affordable space access and speedy global reach. Compared with ordinary aircrafts, AHVs adopt airframe integrated with scramjet engine configuration, which leads to strong interactions among the elastic airframe, the propulsion system, and the structural dynamics. In addition, the requirements of flight stability and high speed response, the existence of various random interference factors and large uncertainties, the design of a robust controller for AHVs using a nonlinear uncertainty model is highly challenging [1,2]. Because of the slender geometries and light structures of this generic vehicle, significant flexible effects cannot be neglected in the controller design [3], since these modes may be harmful to system stability. In order to ensure the safety and reliability, fight control design for AHVs must guarantee stability of the flight system and provide a satisfying control performance [4]. Recently, a FAHV model, which includes the flexible dynamics, was developed in [5–7]. Based on this kind of FAHV models, several studies on the flight controller design and simulation have been published in recent years. The approach of linearizing FAHV model at given operating points and design of a controller using linear control design techniques has been widely used. In [8], the nonlinear longitudinal dynamics of a FAHV was directly linearized at a

n

Corresponding author. E-mail address: [email protected] (J. Wang).

specified trim condition, and a linear quadratic regulator (LQR) was presented for a region in the neighborhood of the operating point. To ensure stability for a wide range of operating points, the idea of feedback linearization attracted considerable attention in [9,10]. The advantage of this method is that it could linearize the nonlinear dynamics by linearizing the input–output map for all values of the states x in a certain neighborhood of x0 instead of the operating point x0 . After obtaining a linearized input–output map, the linear controller can be designed in combination with the feedback linearization law. In [11], approximate feedback linearization was applied to transform the nonlinear FAHV model into a MIMO model which provides an example of nonlinear controller design. In [12,13], a tracking controller was constructed by using a minimax LQR control approach together with feedback linearization law for the nonlinear FAHV model developed by Serrani [14]. In practice, the atmospheric properties and aerodynamic characteristics in the flight envelop of the FAHVs are difficult to be measured or estimated. Therefore, robust control remains a key choice for FAHV flight control. Sliding mode control (SMC) is commonly favored as a powerful robust control method to handle systems running under uncertain conditions [15,16]. However, it is also well known that traditional SMC technique may have the shortcoming of chattering phenomenon in practice [17,18]. A number of methods have been proposed to reliably prevent chattering, such as observer-based solution [16], the boundary layer solution [19], and the adaptive sliding mode method [20]. High order sliding mode (HOSM) control proposed in [21–24] has also been studied in the context of chattering. In [21], the HOSM controller based on homogeneity properties is presented to achieve finite time convergence.

0019-0578/$ - see front matter & 2014 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2014.01.002

J. Wang et al. / ISA Transactions 53 (2014) 690–698

After that, the quasi-continuous HOSM controller is presented in [22], which allow the control practically continuous function with respect to the sliding variable except origin point. In [23], an integral sliding mode approach based on minimization of a quadratic criterion is developed to design a HOSM controller, where the finite time stabilization and uncertainty rejection problem is handled. The robust HOSM controllers for a class of multivariable nonlinear systems proposed in [24] imply that the HOSM control is equivalent to finite time stabilization of a higher order integrator chain system with bounded nonlinear uncertainties. The controller contains a continuous controller guaranteeing finite time stabilization of the nominal system at the origin, and a discontinuous controller enabling to reject the uncertainties. Moreover, this control technique provides a simple constructive condition of the controller gain parameters. Among the above methods, in order to reduce the control chattering, the continuous sliding mode control approach replaces the discontinuous control by a linear saturation control law that is valid within a boundary layer of the sliding manifold. The price of this method is a partial loss of robustness and tracking accuracy to disturbances and model uncertainties. In order to improve the robustness, sliding mode disturbance observer (SMDO) that can estimate the bounded uncertainties is used in the satellite formation control [25]. The SMDO does not rely on complete knowledge of the bounded disturbance mathematical model, just on its bounds, resulting in a simple low-order design. The main contribution of this paper is developing the SMC method for the FAHVs' output to converge the desired command with bounded nonlinear uncertainties via application of continuous HOSM control modulated by Hong's algorithms [26], and then the SMDO developed in [27], which has the high efficiency in accomplishing the nonlinear dynamic estimation, is applied to totally compensate uncertainties. The motivation of the research is to study the tracking control problem for the longitudinal model of a FAHV in the presence of aerodynamic uncertainties. Furthermore, the main contributions of this paper can be summarized as the control strategy combining input–output linearization and continuous HOSM is proposed for the FAHV to guarantee the finite time stability. The key point of this strategy is to design a SMDO for the purpose of handling aerodynamic uncertainties. In addition, the characteristic of HOSM is achieved via homogeneity theory as long as the sliding mode motion occurs. After aerodynamic uncertainties compensation based on SMDO, the good uncertainties rejection and tracking performance of the closed-loop system can be maintained. Finally, simulation results are given to confirm the effectiveness of the proposed approach. The rest of the paper is organized as follows: in Section 2, the problem formulation including flexible hypersonic vehicle model and the Input–Output linearization model followed by controller design is stated. The main results are shown in Section 3, where a continuous HOSM controller is developed by combining the disturbance observer technique to track the desired tracking command for the velocity and altitude of the FAHV despite the model aerodynamic uncertainties. The normal and Monte Carlo tests are conducted for trimmed cruise conditions in altitude 85,000 ft and velocity 7710 ft/s in Section 4, and the conclusions are provided in Section 5.

2. Hypersonic vehicle model The model adopted in our study is originated from the first principles model developed by Bolender and Doman [6,28] for the longitudinal dynamics of a FAHV. The vehicle geometry is shown in Fig. 1.

691

Lf

47 ft

Ln

X canard 1u

40 ft

20 ft

La

X elev

30 ft

33 ft 2

14.4 deg

3deg e c

6.2 deg

1l

xB

Z elev

zB

3.5 ft

Shock

hi

3.5 ft Shear layer

Reflected Shock

Fig. 1. Air-breathing hypersonic vehicle geometry.

This model derived using Lagrange's equations, and the equations of motion, include flexibility effects. The scramjet engine model is taken from Chavez and Schmidt in [29], where the scramjet engines are capable of obtaining oxygen directly from atmosphere rather than carrying it. A simplified model has been derived for controller design and stability analysis by Fiorentini [30,31]. This model approximates the behavior of the first principles model by replacing the aerodynamic and generalized forces and moments with curve-fitted functions of the rigid-body states, the control inputs and the elastic modes. Assuming a flat Earth and normalizing by the span of the vehicle to unit depth, the equations of motion of the longitudinal dynamics are written in the stability axes as V_ ¼ T

cos α  D g m

sin γ

L þ T sin α g  cos γ V mV L þ T sin α g þ Q þ cos γ α_ ¼  V mV Q_ ¼ M=I yy h_ ¼ V sin γ

γ_ ¼

η€ i ¼  2ςm;i ωm;i η_ i  ω2m;i ηi þ N i ; i ¼ 1; 2; 3:

ð1Þ

This model is composed of five rigid-body state variables x ¼ ½V; γ ; α; Q ; hT , in which V; γ ; α; Q ; h are the vehicle speed, flight path angle, angle of attack, pitch rate, and altitude, respectively; ηi , ωm;i and ςm are the generalized flexible coordinate, natural frequencies and damping coefficients of the ith elastic mode. The reader can refer to [31] for a full description of the variables in this model. Because of coupling in aerodynamic forces of the FAHV model (1), some simplifications must be carried out for the purpose of Input–Output linearization. The simplification of the model is necessary because we want to obtain a linearized model, the same simplified process can be found in [32,33]. An Input–Output linearization model is developed by repeated differentiation the output V and h as follows: ::: V ¼ f V þ b11 ϕc þ b12 δe ð2Þ ð4Þ

h

¼ f h þ b21 ϕc þ b22 δe

ð3Þ

where ϕc and δe are control inputs, and the specific expression of f V , f h , b11 , b12 , b21 , and b22 can be computed similarly in [32] of Eqs. (56)–(61). Note that: The accurate expressions of aerodynamic force as lift L, drag D, thrust T, pitching moment M, and the three generalized forces N 1 , N 2 and N 3 are different from [31] and [32]. Compared with [32,33], the main propose of this study is to design control input u ¼ ½ϕc ; δe T based on continuous HOSM controller for the FAHV to follow to track some desired commanded values denoted by yd ¼ ½V d ; hd Τ in the presence of

692

J. Wang et al. / ISA Transactions 53 (2014) 690–698

with respect to the family of dilation δε if q

uncertainties jΔC M j r 0:1C M ; jΔC L jr 0:1C L ; jΔC D j r 0:1C D ; jΔC T jr 0:1C T

ð4Þ

The detailed expressions of aerodynamic coefficients C M ,C L , C D ,C T , and nonlinear coupling uncertain terms ΔC L , ΔC D , ΔC M , and ΔC T can be referred to [32,33]. Note that: In order to verify the effectiveness of the proposed control strategy, the maximum upper bound of uncertainties (4) are added to the aerodynamic force lift L, drag D, thrust T, and pitching moment M.

3. Continuous high order sliding mode controller design According to the tracking command, define the following sliding mode surface as follows: Z t s1 ¼ V  V d þ λV ðV  V d Þdt ð5Þ 0

s2 ¼ h  hd þ λh

Z

t 0

ðh  hd Þdt

ð6Þ

where λV and λh are strictly positive constants, and the integrals of the tracking errors assist greatly in eliminating the steady state errors. Applying the input–output linearization technique [10] to the simplified model, the input–output dynamics can be derived by differentiating s1 three times and s2 four times, we obtain ::: ::: s1 ¼ f V  V d þ λV ðV€  V€ d Þ þb11 ϕc þ b12 δe ð7Þ ::: :::

ð4Þ sð4Þ 2 ¼ f h hd þ λh ð h  hd Þ þb21 ϕc þb22 δe

ð8Þ

Eqs. (7) and (8) can be expressed in matrix form " ::: # " # " #" # " # s1 A1 b11 b12 Δ1 ϕc þ þ ð4Þ ¼ s2 A2 b21 b22 Δ2 δe |fflffl{zfflffl} |fflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflffl}|fflfflffl{zfflfflffl} |fflfflffl{zfflfflffl} B

A ¼ ½A1 ;A2 T

u

ð9Þ

Δ ¼ ½Δ1 ;Δ2 T

where ::: A1 ¼ f V  V d þ λV ðV€  V€ d Þ ð4Þ

ð3Þ

A2 ¼ f h  hd þ λh ðh

ð10Þ

ð3Þ

 hd Þ

ε 40

system z_ ¼ f ðzÞ is called homogeneous if its vector field f is homogeneous, where f ð0Þ ¼ 0, and z ¼ ðz1 ; ⋯; zn ÞΤ . Lemma 1. (Harmouche et al. [34]). Suppose that, the time invariant system (52) below z_ ¼ f ðzÞ; f ð0Þ ¼ 0;

ð13Þ

is homogeneous of degree p with respect to the family of dilations δqε ,f ðzÞ is continuous and z ¼ 0 is its asymptotically stable equilibrium. Then the equilibrium of system (13) is globally finite-time stable. Lemma 2. (Harmouche et al. [34]). Consider system (13). Suppose there is a continuous function VðzÞ : D-R defined on a neighborhood U D D of the origin such that the following conditions hold: 1. V ðzÞ is positive definite function on D D Rn ; 2. There exist real numbers k 4 0 and λ A ð0; 1Þ such that _ þ kV λ ðzÞ r0; z A U=f0g VðzÞ then system (13) is locally finite-time stable. If D ¼ Rn and VðzÞ is also unbounded, system (13) is globally finite-time stable. Assumption 1. The relative degree vector r ¼ ½r1 ; r 2 Τ with respect to s ¼ ½s1 ; s2 Τ is assumed to be constant and known. It means the 2  2 matrix 2 3 Lg1 Lfr1  1 s1 Lg2 Lfr1  1 s1 4 5 ð14Þ B¼ Lg1 Lfr2  1 s2 Lg2 Lfr2  1 s2 is nonsingular. Remark 1. For the Input–Output combination, the matrix B is non-singular over the entire flight envelope given in [31], so the Assumption 1 is reasonable to be assumed.

ð11Þ

note that the additional item Δ is introduced to represent the flexible effects and coupled uncertainties ΔC L ,ΔC D ,ΔC M ,ΔC T and their variations described in Eq. (4). The system (9) with respect to the sliding variable s ¼ ½s1 ; s2 Τ is equivalent to the finite time stabilization of the following multivariable integral chain system: 88 z_ ¼ zi;2 > > < i;1 >> > > ⋮ < 8 i ¼ 1; 2 > : z_ ð12Þ i;r i  1 ¼ zi;r i > > > > > r r Τ : ½s 1 ; s 2  ¼ A þ Bu þ Δ 1 2 ðj  1Þ , i

f j ðεq1 z1 ; …; εqn zn Þ ¼ εp þ qi f j ðzÞ; j ¼ 1; …; n;

with 1 rj r r i , zi;j ¼ s zi ¼ ½zi;1 ; …; zi;ri  . Before giving the control design, we recall some definitions, lemmas, and assumptions which will be utilized in the subsequent control development and analysis.

3.1. Continuous HOSM control design based on disturbance observer Before the SMDO design, a new control variable is introduced as follows: u~ ¼ Bu

ð15Þ

Once the control u~ is identified, the original control u ¼ ½ϕc ; δe T can be easily computed  based  on (15). Let u~ ¼ ½b11 ϕc þ b12 δ T T ¼ ΔAþ ΔBu, then (12) can be e ; b21 ϕc þb22 δe  , and Δ ¼ Δ1 ; Δ2 transformed to the following form ::: T ~ ½s1 ; sð4Þ 2  ¼ AþΔþu

ð16Þ

T

δqε ðz1 ; …; zn Þ ¼ ðεq1 z1 ; …; εqn zn Þ

Theorem 1. Let qi;k ; β i;k  1 ; k ¼ 1; ⋯; r i and p o 0 be constants which satisfy the following given inequalities: 8 qi;1 ¼ 1; …; qi;k ¼ qi;k  1 þ p; qi;k 4 p 4 0; > > > > > k ¼ 1; …; r i  1 < ð17Þ β ¼ q ; ð β þ 1Þq Z ðβi;k  1 þ 1Þqi;k 4 0; > i;2 i;k þ 1 i;k > > i;0 > > k ¼ 1; …; r i  2; βi;ri  1 4 0; :

where z1 ; …; zn are suitable coordinates on Rn and q ¼ ðq1 ; …; qn Þ with the dilation coefficients q1 ; ⋯; qn positive real numbers. A vector field f ðzÞ ¼ ðf 1 ðzÞ; …; f n ðzÞÞΤ is homogeneous of degree p A R

then there exist constants li;k 4 0; k ¼ 1; …; r i  1 such that the control law wi ðzÞ ¼ wi;ri ðzÞ renders the system (12) finite time stable,

Definition 1. (Hong [26]). A family of dilations δε is a mapping that assigns to every real ε 4 0 a diffeomorphism q

J. Wang et al. / ISA Transactions 53 (2014) 690–698

693

where wi ¼ Ai þ u~ i are defined as follows:

Then the time derivative of the Lyapunov function (22) becomes

8 wi;0 ¼ 0 > > > ðqi1 þ pÞ > > > βi;0 βi;0 ðqi1 βi0 Þ > > < wi;1 ¼  li;1 ⌈⌈zi;1 ⌉ ⌈wi;0 ⌉ ⌉

2 ∂W ∂V_ 12 1;3 V_ 13 ¼ V_ 12 ðz1;1 ; z1;2 Þ þ ðz1;3  w1;2 Þ þ ∑ z1;j þ 1 ∂z1;2 j ¼ 1 ∂z1;j |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

⋮ > ðqi;k þ 1 þ pÞ > > > βi;k  ⌈w ⌉βi;k ⌉ðqi;k þ 1 βi;k Þ > w ¼  l ⌈⌈z ⌉ > i;k þ 1 i;k þ 1 i;k þ 1 i;k > > : k ¼ 1; …; r  1 i

V 13 0

ð18Þ þ

when z1;3 ¼ w1;2 , we have W 1;3 ¼ 0, and therefore

we denote ⌈x⌉κ ¼ jxjκ signðxÞ, where κ is a positive real number. Proof. We proceed to prove the Theorem 1 by induction based on systems (12) with i ¼ 1; j ¼ 1; 2; 3 for instance. According to the aforementioned analysis, the closed-loop system is homogeneous q of degree p o 0 with respect to the family of dilations δε for q ¼ ðq1;1 ; q1;2 ; q1;3 Þ. According to Lemma 1; system (19) below 8 > z_ 1;1 ¼ z1;2 < z_ 1;2 ¼ z1;3 ð19Þ > : z_ ¼ w 1;3 1 with z1;j ¼ sðj1  1Þ , j ¼ 1; 2; 3 under the feedback control law w1 is globally finite-time stable once it is asymptotically stable. The asymptotic stability can be proved together using the induction method. Step 1: Whenj ¼ 1, consider z_ 1;1 ¼ w1 , for any l1;1 4 0, w1 can be obtained as follows:

V_ 13 ¼ V_ 12 ðz1;1 ; z1;2 Þ o 0 if z1;3 a w1;2 , without loss of generality, the study can be restricted to the unit sphere s1 ¼ fz1 A R3 : Γ 3 ¼ 1g, Γ 3 is defined as

Γ 3 ¼ ðjz1;1 jc=q1;1 þ jz1;2 jc=q1;2 þ jz1;3 jc=q1;3 Þ1=c where c 4 max fq1;1 ; q1;2 ; q1;3 g.. Inspired by [26], we defines1 þ ¼ fz1 A R3 : V 13 0 Z 0g, s1  ¼ fz1 A R3 : V 13 0 o 0g. If s1 þ in empty, then V 13 0 o 0 according to the expression ðq1;3 þ pÞ

w1 ¼ w1;3 ¼  l1;3 ⌈⌈z1;3 ⌉β1;2  ⌈w1;2 ⌉β1;2 ⌉ðq1;3 β1;2 Þ V_ 13 is negative,which can be expressed as the following inequality: V_ 13 ¼ V_ 12 ðz1;1 ; z1;2 Þ þ V 13 0 ðq1;3 þ pÞ

 l1;3 j⌈z1;3 ⌉β1;2  ⌈w1;2 ⌉β1;2 jðq1;3 β1;2 Þ next, if s1

n

þ

þ1

o0

\ s1





¼  l1;1 ⌈z1;1 ⌉ðq1;1 þ pÞ=q1;1

M2 ¼

Take the Lyapunov function as ð20Þ

then the time derivative of the Lyapunov function (20) becomes

min

z1 A s1 \ s1 þ

ðq1;3 þ pÞ

j⌈z1;3 ⌉β1;2  ⌈w1;2 ⌉β1;2 jðq1;3 β1;2 Þ

þ1



ðq1;3 þ pÞ

þ1 V_ 13 ¼ V_ 12 ðz1;1 ; z1;2 Þ þ V 13 0  l1;3 j⌈z1;3 ⌉β1;2  ⌈w1;2 ⌉β1;2 jðq1;3 β1;2 Þ

ð24Þ

consider 0 a z1 A R3 , and define e as

¼  l1;1 ð1 þ q1;2 Þ⌈z1;1 ⌉q1;2 U⌈z1;1 ⌉ðqi1 þ pÞ=qi1 ¼  l1;1 ð1 þ q1;2 Þjz1;1 j

a ∅, denote

when selecting l1;3 4M 1 =M 2 , (24) can be rewritten as

oM 1  l1;3 M 2 o 0

V_ 11 ¼ ð1 þ q1;2 Þ⌈z1;1 ⌉q1;2 z_ 1;1 q1;2 þ 1 þ p

ð23Þ

z1 A s1

ðq1;1 þ pÞ

ð21Þ

¼  l1;1 ð1 þ q1;2 Þðjz1;1 j1 þ q1;2 Þðq1;2 þ 1 þ pÞ=ð1 þ q1;2 Þ

e ¼ ðz1;1 =Γ 3

q1;1

; z1;2 =Γ 3

q1;2

; z1;3 =Γ 3

q1;3 Τ

Þ A s1

then, for any z1 a 0 due to homogeneity, V_ 13 can be rewritten as

¼  l1;1 ð1 þ q1;2 ÞV 11 ðq1;2 þ 1 þ pÞ=ð1 þ q1;2 Þ denote λ1;1 ¼ ðq1;2 þ 1 þpÞ=ð1 þ q1;2 Þ A ð0; 1Þ, then (21) can be rewritten as V_ 11 ¼  μ1;1 V 11 λ1;1 where μ1;1 ¼ l1;1 ð1 þq1;2 Þ. Therefore, according to the Lemma 2, w1 stabilizes the system z_ 1;1 ¼ w1 in finite time. Step 2: Assume that the conclusion holds true when j ¼ 2. In other words, its Lyapunov function V 12 is positive definite with respect to z1;1 and z1;2 , and moreover, its derivative V_ 12 is negative definite. When j ¼ 3, we define W 1;3 as h i Rz W 1;3 ¼ ⌈w1;31;2 ⌉ ⌈s⌉β1;2 ⌈w1;2 ⌉β1;2 ds 1 ¼ ½jz jðβ1;2 þ 1Þ þ β1;2 jw1;2 jðβ1;2 þ 1Þ   z1;3 ⌈w1;2 ⌉β1;2 β1;2 þ 1 1;3 it can be seen that W 1;3 is a positive definite function when z1;3 a w1;2 . Consider the system (19), and define a Lyapunov function V 13 as follows: V 13 ¼ V 12 þ W 1;3

þ

is nonempty, then s1 o M 1 ¼ max V 13 0

w1 ¼ w1;1 ¼  l1;1 ⌈⌈z1;1 ⌉β1;0 ⌈w1;0 ⌉β1;0 ⌉ðq1;1 β1;0 Þ

V 11 ¼ jz1;1 j1 þ q1;2

∂W 1;3 z_ 1;3 ∂z1;3

ð22Þ

ðβ þ 1Þq1;3 þ p _ V_ 13 ðz1 Þ ¼ Γ 3 1;2 V 13 ðeÞ o 0

consequently, system (19) is globally finite-time stable based on Lemma 1. Similarly to (19), the following inequality can be obtained 8 _ z2;1 ¼ z2;2 > > > > < z_ 2;2 ¼ z2;3 ð25Þ z_ 2;3 ¼ z2;4 > > > > : z_ ¼ w 2;4

2

With z2;j ¼ sðj2  1Þ , j ¼ 1; 2; 3; 4 under the feedback control law w2 ðz2 Þ ¼ w2;r2 ðz2 Þ is globally finite-time stable. Theorem 1. provides the control of an integrator chain system, whereas the real problem to be tackled is the control of system (16) with uncertainties. The sliding variable dynamics (16) is sensitive to the unknown bounded item Δ ¼ ½Δ1 ; Δ2 Τ . In order to address the issue, the disturbance observer is proposed to estimate and cancel the uncertainties. ::: € 1 ðxÞ; s2 ðxÞΤ and u~ ¼ ½u~ 1 ; u~ 2 Τ be availLet the variables S ¼ ½s ðn  1Þ able in real time, and Δ be n  1 times differentiable, so that Δ has a known Lipshitz constant L 4 0. The control function u~ is Lebesgue-measurable. The proposed sliding mode disturbance

694

J. Wang et al. / ISA Transactions 53 (2014) 690–698

This observer has a slightly different structure from the one in [27] because a relative degree two system is considered in (26). Whereas in the work of [27] relative degree one structures are tackled.

observer structure can be written as follows: z_ i;0 ¼ vi;0 þ u~ i vi;0 ¼  λ0 Ln þ 1 jzi;0 Si jn þ 1 signðzi;0  Si Þ þ zi;1 1

n

⋮ z_ i;n  1 ¼ vi;n  1 vi;n  1 ¼  λn  1 L jzi;n  1  vi;n  2 j signðzi;n  1  vi;n  2 Þ þ zi;n 1 2

1 2

z_ i;n ¼  λn L signðzi;n  vi;n  1 Þ

ð26Þ

In particular, in the absence of input noises the exact equalities are established in a finite time ðj  1Þ zi;0 ¼ u~ i ; zi;1 ¼ Δi ; …; zi;j ¼ Δi ; j ¼ 1; 2; …; n

ð27Þ

Corollary 1. Let the sliding variable dynamics be of the form (16), with Δi being n  1 smooth with a known Lipshitz constant L of Δði n  1Þ . Based on Theorem 1, the continuous HOSM controller can be designed as follows: u~ i ¼  Ai  zi;1  wi ð28Þ when exact measurements are available, zi;1 becomes equal to Δi in a finite time. Then, finite time stable system (17) is achieved via continuous HOSM control. Based on (15), the actual control input

Table 1 Initial and desired final conditions for the state variables and control inputs.

V

Initial value (trim value) 7710 ft=s

Final trim

Flexibles states and control inputs

7810 ft=s

η1

85000 ft

87000 ft

γ

0 deg

0 deg

η3

α Q

1:7373 deg 0 deg=s

1:6551 deg 0 deg=s

ϕc δe

Velocity Tracking [ft/s]

h

η2

7850 7800 7750

Vd V

7700 0

50

100

150

200

250

Initial value (trim value) 0:50277 ft slug

Altitude Tracking [ft]

Variables

8.8

[deg]

c

e

control input

control input

0.35 0.3 0.25 150

200

250

 0:012692 ft slug 0:094715 5:2221 deg

 0:0038 ft slug 0:2827 5:4126 deg

1=2

hd h

8.4 50

100

150

200

250

300

200

250

300

200

250

300

200

250

300

6 5.8 5.6 5.4 5.2

300

0

50

100

150

Time [s]

1.9

0.15

1.8

0.1

Q [deg/s]

[deg]

1=2

8.5

Time [s]

1.7

0.05 0

1.6 1.5

-0.05 0

50

100

150

200

250

300

0

50

100

Time [s]

150

Time [s]

-7

0.6 0.4

[deg]

-7.5

0.2

c

[deg]

 0:0611 ft slug

1=2

8.6

0

-8

0

-8.5

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Fig. 2. Response to 100 ft/s step velocity command and 2000 ft step altitude command. (a) Regulated outputs and control inputs (b) Other flight states.

J. Wang et al. / ISA Transactions 53 (2014) 690–698

0.2

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Fig. 4. Sliding mode surfaces1 , s2 and their derivatives.

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Fig. 5. Response to 100 ft/s step velocity command and 2000 ft step altitude command under 200 times Monte-Carlo Simulation. (c) Regulated outputs and control inputs (d) Other flight states.

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J. Wang et al. / ISA Transactions 53 (2014) 690–698

can be expressed as follows: " # A1  z1;1  w1 1 u¼B A2  z2;1  w2

ð29Þ

Despite the sign function included in (28), the item ⌈x⌉κ ¼ jxjκ signðxÞ is a continuous function [26]. Furthermore, according to the Assumption 1, we can see that the matrix B is non-singular. Therefore, the control input (29) is available as long as we obtain (28). Remark 2. The main advantage of our research can be summarized: the proposed continuous HOSM compare with the traditional sliding mode control, such as finite-time stability, and the reduced control chattering [20,25]. Meanwhile, in order to improve the robustness, SMDO is applied for the FAHV model controller design to estimate the bounded uncertainties without complete knowledge of the uncertainties upper bound. 4. Simulation results

Case 1. Normal test In the normal case, the velocity V is generated to accelerate the vehicle to the desired tracking command V d ¼ 100 ft=s in 150 s, and the altitude h is generated to control the vehicle to climb the desired tracking command hd ¼ 2000 ft in 100 s. It can be seen from the results of the simulation that the proposed controller provides stable tracking of the reference trajectories and makes the system

7850

8.8

Altitude Tracking [ft]

Velocity Tracking [ft/s]

To illustrate the efficiency of controller designed previously, a climbing maneuver with longitudinal acceleration for a 100 ft/s

velocity change and a 2000 ft altitude change is considered. Simulation studies have been done on the full nonlinear flexible hypersonic vehicle described in previous. The initial and final trim conditions for each case are listed in Table 1. In the simulations, the reference commands have been generated by filtering step reference commands through a second-order pre-filter with natural frequency ωf ¼ 0:06 rad=s and damping factor ζ f ¼ 0:95. The vehicle is initially at trim, and the SMDO parameters are chosen as L ¼ 3, λ0 ¼ 1:5, λ1 ¼ 1:3, and λ2 ¼ 1:1. The parameter p is taken as p ¼ 0:1. The control parameters are selected as lV1 ¼ 0:5, lV 2 ¼ 0:5, lV3 ¼ 5, lh1 ¼ 0:2, lh2 ¼ 0:3, lh3 ¼ 0:5, and lh4 ¼ 7. Two cases studied based on continuous HOSM controller together with SMDO are shown as follows:

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Fig. 6. Response to 100 ft/s step velocity command and 2000 ft step altitude command with traditional SMC proposed in [2]. (e) Regulated outputs and control inputs (f) Other flight states.

J. Wang et al. / ISA Transactions 53 (2014) 690–698

converge to the desired trim condition. Specifically, the tracking performance for the altitude and velocity change is shown in the top plots of Fig. 2(a). The tracking error remains remarkably small during the entire maneuver and vanishes asymptotically. The control inputs of fuel equivalence ratio ϕc and elevator deflection δe settle down with no chattering which could be seen in bottom plots of Fig. 2(a). Fig. 2(b) shows the performance of the angle of attack α and the pitch rate Q at the top, as well as the canard deflection δc and the flight path angle γ at the bottom. Furthermore, the six flexible states are provided in Fig. 3 which means the flexible states converge to trimmed points after a short time. Meanwhile, the velocity and altitude sliding mode surface s1 , s2 as well as the corresponding derivatives are given in Fig. 4. It can be seen that ::: s1 ; s_ 1 ; s€ 1 and s2 ; s_ 2 ; s€ 2 ; s2 converge to zero after in a finite time which verify the effectiveness of the proposed control strategy. From the simulation results, it can be seen that the proposed continuous HOSM controller together with SMDO provides good tracking performance for the full nonlinear flexible hypersonic vehicle. Case 2. Monte-Carlo test In order to verify the effectiveness and robustness of the controller, the 200 times Monte-Carlo simulation with the uncertainties provided in (4) is carried out in this section. The main flight states and control inputs information are plotted in Fig. 5. In the top plots of Fig. 5(c), it can be seen that the velocity and altitude can track the desired commands in spite of the uncertainties, and the control inputs fuel equivalence ratio ϕc and elevator deflection δe are bounded in bottom plots of Fig. 5(c). The angle of attack α is shown in Fig. 5(d). According to these figures, we can see that the angle of attack has different steady-state values due to the existence of the uncertainties defined in (4). For example, the angle of attack has to be increased when the drag force coefficient has positive uncertainty and lift force coefficient has negative uncertainty. Whereas the steady-state value of flight path angle γ is not affected by the aerodynamic uncertainties which can be verified from Fig. 5(d). Finally, the simulation results shown in Fig. 5 demonstrate that the proposed control strategy is effective in stable tracking performance and exhibits robustness with respect to aerodynamic uncertainties. In the aforementioned two cases, the velocity and altitude converge to the desired commands in finite time and there are no steady-state errors for the FAHV model. In general, it is hard to find a controller with good robustness properties and performance at the same time. As it can be seen from Figs. 2–4, the simulation results illustrate that the proposed controller can stabilize the system and guarantee satisfactory tracking performance with good robustness properties simultaneously. Case 3. Comparison results In order to demonstrate the effectiveness of the proposed composite control strategy (29), the traditional sliding mode controller proposed in [2] is used for comparison. As presented in Fig. 6(e) and (f), the traditional SMC proposed in [2] works effectively and achieves the control target in the presence of uncertainties. However, this approach faces an unavoidable application problem: chattering, which is presented in depict curves of the control inputs of fuel equivalence ratio ϕc and elevator deflection δe settle down in bottom plots of Fig. 6(e). From the comparison results we can conclude the effectiveness of proposed composite control method, continuous HOSM controller is applied to reducing the control chattering, disturbance observer technique is employed to estimate the uncertainties and compensate the control inputs through the estimated value. Fig. 6(f) describes the curve of other flight states.

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5. Conclusions In this paper, the combination of continuous HOSM and SMDO for the longitudinal dynamics of the FAHV is proposed to track the desired tracking commands in spite of aerodynamic uncertainties. The continuous HOSM controller together with SMDO computed a robust continuous control which is able to compensate for the uncertainties effectively, and Monte-Carlo simulation verifies the effectiveness of the proposed strategy.

Acknowledgments This research was supported in part by National Natural Science Foundation of China (Nos. 61273092, and 61203012), the Foundation for Key Program of Ministry of Education, China (No. 311012), the Key Program for basic research of Tianjin (Grant No. 11JCZDJC25100), the Key Program of Tianjin Natural Science (No. 12JCZDJC30300). Aeronautical Science Foundation of China (No. 20125848004) Supported by Science and Technology on Aircraft Control Laboratory. References [1] Wang Q, Stengel RF. Robust nonlinear control of a hypersonic aircraft[J]. J Guidance Control Dyn 2000;23(4):577–85. [2] Xu HJ, Mirmirani MD, Ioannou PA. Adaptive sliding mode control design for a hypersonic flight vehicle[J]. J Guidance Control Dyn 2004;27(5):829–38. [3] Bolender MA, Doman DB. A non-linear model for the longitudinal dynamics of a hypersonic air-breathing vehicle[C]. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference; 2005: AIAA-6255. [4] Wilcox Z, MacKunis W, Bhat S, et al. Robust nonlinear control of a hypersonic aircraft in the presence of aerothermoelastic effects[C]. American Control Conference, Missouri, USA; June, 2009: pp. 2533–2538. [5] Williams T, Bolender MA, Doman DB, et al. An aerothermal flexible mode analysis of a hypersonic vehicle[C]. AIAA Atmospheric Flight Mechanics Conference and Exhibit; 2006, AIAA Paper 2006-6647. [6] Bolender MA, Doman DBA. Nonlinear longitudinal dynamical model of an airbreathing hypersonic vehicle[J]. J Spacecr Rockets 2007;44(2):374–87. [7] Bolender MA. An overview on dynamics and controls modeling of hypersonic vehicles[C]. In: Proceedings of the 2009 American Control Conference, St. Louis, MO, USA; 2009: pp. 2507–2512. [8] Groves KP, Sigthorsson DO, Serrani A, et al. Reference command tracking for a linearized model of an air-breathing hypersonic vehicle[C]. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference; 2005: AIAA-6144. [9] Brockett RW. Feedback invariants for nonlinear systems[C]. In: Proceeding of IFAC World Congress, Helsinki, Finland; 1978: pp. 1115–1120. [10] Isidori A. Nonlinear Control Systems. 3rd ed.. London, UK: Springer; 1995 ([M]). [11] Parker JT, Serrani A, Yurkovich S, et al. Control-oriented modeling of an airbreathing hypersonic vehicle[J]. J Guidance Control Dyn 2007;30(3):856–69. [12] Rehman O, Fidan B, Petersen RI. Uncertainty modeling for robust minimax LQR control of hypersonic flight vehicles[C]. In: Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit; 2010: AIAA-8285. [13] Rehman O, Petersen IR, Fidan B. Robust nonlinear control design of a nonlinear uncertain system with input coupling and its application to hypersonic flight vehicles[C]. In: Proceedings of the 2010 IEEE International Conference on Control Applications – part of 2010 IEEE Multi-Conference on Systems and Control, Yokohama, Japan; 2010: pp. 1451–1457. [14] Sigthorsson D.O., Serrani A. Development of linear parameter-varying models of hypersonic air-breathing vehicles[C]. AIAA Guidance, Navigation and Control Conference, 2009, AIAA-6282. [15] Wang T, Xie WF, Zhang YM. Sliding mode fault tolerant control dealing with modeling uncertainties and actuator faults [J]. ISA Trans 2012;51(3):386–92. [16] Utkin VI, Guldner J, Shi J. Sliding Mode in Control in Electromechanical Systems[J]. London: Taylor & Francis; 1999. [17] Boiko I, Fridman L. Analysis of chattering in continuous sliding-mode controllers [J]. IEEE Trans Autom Control 2005;Vol.50:1442–6. [18] Xu YJ. Chattering free robust control for nonlinear systems[J]. IEEE Trans Control Syst Technol 2008;16:1352–9. [19] Slotine JJ. Sliding mode controller design for non-linear systems[J]. Int J Control 1984;40:421–34. [20] Mondal S, Mahanta C. A fast converging robust controller using adaptive second order sliding mode [J]. ISA Trans 2012;51(6):713–21. [21] Levant A. Homogeneity approach to high-order sliding mode design [J]. Automatica 2005;41(5):823–30. [22] Levant A. Quasi-continuous high-order sliding-mode controllers[J]. IEEE Trans Autom Control 2005;50(11):1812–6.

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