Nonlinear robust control for reusable launch vehicles in reentry phase based on time-varying high order sliding mode

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Journal of the Franklin Institute 350 (2013) 1787–1807 www.elsevier.com/locate/jfranklin

Nonlinear robust control for reusable launch vehicles in reentry phase based on time-varying high order sliding mode Bailing Tiana,n, Wenru Fanb, Qun Zonga, Jie Wanga, Fang Wanga a

The Tianjin Key Laboratory of Process Measurement and Control, School of Electric and Automation Engineering, Tianjin University, Tianjin 300072, China b Aeronautical Automation College, Civil Aviation University of China, Tianjin 300300, China Received 21 November 2012; received in revised form 2 April 2013; accepted 25 April 2013 Available online 10 May 2013

Abstract This paper describes the design of nonlinear robust controller for reusable launch vehicles which is nonlinear, multivariable, strong coupling, and includes uncertain parameters. Firstly, the feasible guidance strategy is proposed to obtain the desired guidance commands. Then, the time-varying sliding mode manifold is designed through calculating a series of algebraic equations with fixed final states to make the system trajectories start on the manifold at the initial time. The global robustness is ensured via designing high order sliding mode attitude controller which forces the system trajectory to stay on the sliding mode manifold despite the model parameter uncertainties and external disturbances. Furthermore, in order to reduce control saltation, the virtual control is introduced into the control strategy. Finally, the six degree of freedom flight simulation results are provided to demonstrate the effectiveness of the integrated guidance and control strategy in tracking the guidance commands as well as achieving safe and stable reentry flight. & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction As a result of technological progress, we are now on verge of developing cost-effective reusable launch vehicles (RLV) for space access. A RLV refers to the vehicle that can be launched into space more than once. Due to the fact that they can be re-used, RLV will dramatically reduce the cost of access to space. However, the technical challenges of designing a n

Corresponding author. Tel.: +86 22 27892382. E-mail address: [email protected] (B. Tian).

0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfranklin.2013.04.022

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control system to fly to orbit and return are monumental, especially in reentry phase. This is due to the fact that there are enormous amounts of model parameter uncertainties, external disturbances and complex path constraints such as heat rate, structural loads and dynamic pressure during this phase. All the factors should be handled carefully for the purpose of safe and stable reentry flight [1–3]. Advanced guidance and control (AG&C) technology is critical for meeting safety, reliability, and cost requirements for the next generation of RLV. And these methods have received considerable attention during the last decades. The main task of the entry guidance is on-board correction of the flight path to compensate for disturbances and model uncertainties. An effective reentry guidance law has been designed that computes the commanded angle of attack and bank angle [4]. Next, the obvious emphasis should be on tracking the guidance commands by designing an appropriate attitude controller which is also the focus of the research. The attitude controller should track the guidance commands while at the same time should have sufficient robustness for external disturbances and inaccuracies in the system model. Conventionally, gain scheduling is primary control method employed on flight control [5,6]. Using the method, the flight control design is carried out by linearizing the system at a series of operating points and designing separate controllers at each of these points. Finally, the overall flight control system is realized in the philosophy of gain scheduling where the individual gains are interpolated with respect to some meaningful system parameters such as dynamic pressure and Mach number, etc. Although gain scheduling is proved to be a powerful and successful method, but has a distinct drawback for RLV application [7]: the number of required gains to be designed and scheduled within the controller is very large. If one also imposes the design constraint that these gains have to allow for a range of possible missions, payloads, and anticipated failure modes, then the number of required gains becomes prohibitive. In addition, the method involves the lack of guaranteed global robustness and stability [8]. Many advanced flight control methods based on nonlinear and/or time varying theory have been developed for the RLV’s attitude control in order to improve the performance of gain scheduling. Trajectory linearization control (TLC) [9,10] as a novel nonlinear control method is used in RLV attitude control. TLC combines an open-loop nonlinear dynamic inversion and a linear time varying (LTV) feedback stabilization computed using parallel-differential eigenstructure assignment [11]. TLC can achieve exponential stability along the nominal trajectory, therefore it provides robust stability and good tracking performance without interpolation of control gains. However, since the trajectory linearization controller linearizes the tracking dynamics, the robustness of the method is limited. Dynamic inversion (DI) technique [12,13] has also been applied in RLV control. DI is a technique for control law design in which feedback is used to simultaneously cancel system dynamics and achieve desired dynamic response characteristics. However, DI control laws lack robustness for modeling errors and parameter inaccuracies if improperly designed. Johnson [14] proposed an adaptive control law to avoid the drawback of DI and to improve the flight control performance. The method used neural network (NN) to compensate for the errors induced from DI. The simulation results illustrate that the adaptive controller has performed well throughout a range of flight conditions. Another nonlinear control method is state dependent Riccati equation (SDRE) [15,16]. This strategy is well-known and has become very popular within the control community over the last decade, providing an effective algorithm for synthesizing nonlinear feedback control by allowing nonlinearities in the system states while additionally offering great design flexibility through state-dependent weighting matrices. However, SDRE needs online computations of algebraic

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Riccati equation at each sample state. It may pose an implementation problem if the system order is higher. Recently, Lam and Xin and others [17–19] proposed the Theta-D control method. Through making modifications to a quadratic performance cost function, it obtains a closed-form solution for the Theta-D control law that is easy to implement and obviates a large amount of online computations. However, this technique suffers from the problem of sub-optimality which means the solution may be far away from the optimal solution. Shtessel and Hall [20–22] have investigated the application of sliding mode to RLV attitude control as a means of achieving both stability and robustness. A discontinuous bang–bang-type control is used to move the system’s states to the sliding mode manifold and rapid switching of that control is used to keep it on the manifold thereafter. In the above methods, in order to reduce the control chattering, the discontinuous sign function is replaced by a boundary layer function. 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 provides estimates of the bounded disturbances/uncertainties is used in the satellite formation control [24]. The SMDO does not rely on complete knowledge of the bounded disturbance mathematical model but just on its bounds. Subsequently, to improve the high-gain switching control because of the conservative estimation of disturbance bounds, Hall and Shtessel [25] combine the gain adaptation algorithm with the SMDO to provide the least gain needed for RLV’s flight control. Furthermore, to reduce control chattering, the high order SMC (HOSMC) is used to launch vehicle attitude control [26]. Preserving the SMC features, the HOSMC technique is capable of removing both the chattering and the relative-degree restrictions and improving its accuracy. There are many papers available on second-order sliding mode [27–29]. An arbitrary order sliding mode controller has been proposed by Levant et al. [30–33], Laghrouche et al. [34,35] and Defoort et al. [36]. At the same time, the time-varying sliding mode control has also been used in flight control. The fault tolerance for RLV attitude control is achieved through a time varying SMC (TVSMC) [23]. To ensure the global robustness, the exponential TVSMC is proposed by Cong [37]. Furthermore, combining the TVSMC and HOSM, the time-varying HOSMC is designed by Plestan [38]. It has two main advantages: the convergence time is precisely known a priori and the sliding mode occurs as early as the initial time, which ensures robustness all over the entire response of the system. However, the sliding manifold parameters have relationship with convergent time which means that these parameters have to be calculated repeatedly when the convergent time changes. In addition, these parameters are obtained via calculating matrix exponential which is timeconsuming and not suitable for fast tracking in RLV attitude especially when the convergent time has to change during the flight. Within the framework of 6-DOF integrated guidance and control for RLV, this paper is part of an effort in achieving the attitude tracking with the convergent time known in advance. The main characteristics of the proposed controller are as follows: the convergence time is precisely known in advance; the sliding mode parameters are independent of the convergence time and are determined only by the order of the system which means the convergence time can be adjusted easily without complex calculation; the sliding mode occurs as early as the initial time which ensures the robustness over the entire response of the system. The paper is organized as follows: the problem studied in the research is formulated in Section 2. In Section 3, the time-varying sliding mode manifold and corresponding controller is designed to make the system output track the guidance command with the convergent time known in advance. The 6-DOF integrated guidance and control simulation results are analyzed in Section 4. The conclusions and further work are presented in Section 5.

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2. Problem formulation 2.1. Six-degree-of-freedom equations of motion The equations for six-degree-of-freedom unpowered rigid flight vehicle are described in Ref. [39]. The motion of the six-degree-of-freedom vehicle can be separated into the motion of the center-of-mass (translational motion), and the motion about the center-of-mass (rotational or attitude motion). 2.1.1. Translational equations of motion The center-of-mass motion is caused by the forces that act on the vehicle and is used to describe the flight trajectory. During the entry phase, most applications assume steady, coordinated turns such that the sideslip angle is kept to zero. Thus, translational motion for reentry flight is given by h_ ¼ v sinγ

ð1Þ

v cosγ sinχ ϕ_ ¼ ðRE þ hÞ cosθ

ð2Þ

θ_ ¼

v cosγ cosχ ðRE þ hÞ

D −g sinγ þ Ω2 ðRe þ hÞ cosθ ðsinγ cosθ−cosγ sinθ cosχÞ m
 L coss g v − − γ_ ¼ cosγ þ 2Ω cosθ sinχ mv v ðRE þ hÞ Ω2 ðRE þ hÞ cosθðcosγ cosθ þ sinγ sinθ cosχÞ þ v

v_ ¼ −

χ_ ¼

ð3Þ ð4Þ

ð5Þ

L sins v Ω2 ðRE þ hÞ þ sinθ cosθ sinχ cosγ sinχ tanθ−2Ωðtanγ cosθ cosχ−sinθÞ þ mv cosγ ðRE þ hÞ v cosγ

ð6Þ where vehicle parameters are as follows: h is the altitude; v is the velocity; ϕ is the latitude; θ is the longitude; γ is the flight path angle; χ is the heading angle; L is the lift force; D is the drag force; g is the gravity (g ¼ μ=ðRE þ hÞ2 with μ being the Earth gravity constant); Ω is the Earth angular speed; RE is the radius of Earth. The control input used for trajectory optimization and guidance law design are angle of attack α and the bank angle s. 2.1.2. Rotational equations of motion The rotational motion is caused by the moments about the center-of-of mass that act on the vehicle and is used to design the altitude controller during the entry flight, which can be given as follows: p_ ¼

ðI yy −I zz ÞI zz −I 2xz I zz M x I xz M z ðI xx −I yy þ I zz ÞI xz þ þ pq þ qr I xx I zz −I 2xz I xx I zz −I 2xz I xx I zz −I 2xz I xx I zz −I 2xz

ð7Þ

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q_ ¼

M y I xz 2 2 I zz −I xx þ ðr −p Þ þ pr I yy I yy I yy

ð8Þ

r_ ¼

ðI xx −I yy ÞI xx þ I 2xz I xz M x I xx M z ðI yy −I xx −I zz ÞI xz þ þ pq þ qr 2 2 2 I xx I zz −I xz I xx I zz −I xz I xx I zz −I xz I xx I zz −I 2xz

ð9Þ

sinðsÞ  χ_ cosðγÞ−ϕ_ sinðχÞ sinðγÞ þ ðθ_ þ ΩÞ cosðβÞ  cosðsÞ  _ γ_ −ϕ cosðχÞ−ðθ_ þ ΩÞ cosðϕÞ sinðχÞ ðcosðϕÞ cosðχÞ sinðγÞ−sinðϕÞ cosðγÞÞ− cosðβÞ ð10Þ

α_ ¼ −p cosðαÞ tanðβÞ þ q−r sinðαÞ tanðβÞ þ

β_ ¼ p sinðαÞ−r cosðαÞ þ sinðsÞ½_γ −ϕ_ cosðχÞ þ ðθ_ þ ΩÞ cosðϕÞ sinðχÞ þcosðsÞ½_χ cosðγÞ−ϕ_ sinðχÞ sinðγÞ−ðθ_ þ ΩÞðcosðϕÞ cosðχÞ sinðγÞ−sinðϕÞ cosðγÞÞ

ð11Þ

s_ ¼ −p cosðαÞ cosðβÞ−q sinðβÞ−r sinðαÞ cosðβÞ þ α_ sinðβÞ−_χ sinðγÞ−ϕ_ sinðχÞ cosðγÞ þðθ_ þ ΩÞ½cosðϕÞ cosðχÞ cosðγÞ þ sinðϕÞ sinðγÞ

ð12Þ

where p, q, r denote roll, pitch and yaw angular rate, respectively;β denotes sideslip angle; I ij ði ¼ x; y; z; j ¼ x; y; zÞ denote moments inertia; M x , M y , M z are roll, pitch and yaw moments. 2.2. Mission analysis 2.2.1. Control objective The whole reentry flight is mainly composed of three parts: reentry phase (AB), terminal area energy management-TAEM (BC) phase and auto landing phase (CD) as depicted in Fig. 1. The flight parameters used in translational and rotational equations are given in Fig. 2. The research is focused on the reentry phase and the objective is to make the vehicle fly from point A to point B in the presence of uncertainties, external disturbances and path constraints. To achieve the goal, it involves determining a flight path to take, guidance system synthesis to follow the desired

Fig. 1. RLV reentry flight.

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Fig. 2. Nomenclature used in RLV reentry.

Trajectory Design

Guidance Law Design

Attitude Controller Design

Mission Objective (Cost & Constraints)

Trajectory Design (Off line, Open loop)

Disturbances

Guidance Law (On line, Close loop)

Attitude Controller

6-DOF RLV Vehicle

(Uncertainties & Coupling)

Fig. 3. Integrated guidance and control architecture for RLV.

trajectory in the presence of disturbances and errors in the models, and maintenance of an attitude profile needed to follow that path. 2.2.2. Entry trajectory constraints During the RLV reentry phase, the flight trajectory must satisfy some typical inequality constraints including heating rate, dynamic pressure and load factor: _ ¼ ðh0 þ h1 α þ h2 α2 þ h3 α3 ÞCρN vM ≤Q _ max Q ð13Þ q ¼ 0:5ρv2 ≤qmax pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 þ D2 ≤nmax

ð14Þ ð15Þ

where Eq. (13) is the heating rate at a stagnation point on the surface of the vehicle. The peak _ max of the heating rate depends on the material of the thermal protection system allowable value Q and the location of the point on the vehicle. There may be multiple such constraints for a number

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of strategic points. But for simplicity, we only consider one such constraint in this research. The constraint in Eq. (14) is dynamic pressure. This constraint may be required for vehicles with movable aerodynamic control surfaces, where the hinge moments are proportional to the dynamic pressure, and must be kept in check. The constraint in Eq. (15) is the total load factor. This constraint is usually decided by considerations in structural stress, particularly for vehicles with lifting body configurations. All the preceding constraints are considered “hard” constraints in that they should be observed with reasonably tight tolerance. 3. Integrated guidance and control strategy The integrated guidance and control strategy includes three parts: ① trajectory design: design reentry trajectory prior to flight to make it satisfy all the path constraints. ② Guidance law design: design guidance law to track the desired trajectory in the presence of uncertainties. ③ Attitude controller design: design attitude controller to maintain the vehicle attitude commanded generated by the guidance program. The connections between trajectory, guidance and attitude control are synthesized in integrated guidance and control architecture for 6-DOF RLV (see Fig. 3). The implementations for each part are summarized as follows. 3.1. Reference reentry trajectory design Since the sideslip angle should be kept to zero to prevent excessive heat buildup, the primary control variables in trajectory optimization are angle of attack and the bank angle. Take maximum cross range reentry trajectory as an example; the objective of trajectory design is equivalent to finding control history uðtÞ ¼ ½αðtÞ; sðtÞT that minimizes the final latitude J ¼ θðt f Þ

ð16Þ

subject to the dynamic model of the preceding section, the initial conditions: h ¼ 260; 000 ft, v ¼ 24; 061 ft/s, ϕ ¼ 01, θ ¼ 01, γ ¼ −1:0641, χ ¼ 01; the final conditions (TAEM interface) h ¼ 80; 000 ft, v ¼ 25; 000 ft/s and γ ¼ −51. In the simulation, the maximum value for path _ max ¼ 200 Btu/ft2/s, qmax ¼ 280 slug/ft2, nmax ¼ 2:5. An exponential atmospheric constraints is: Q model ρ ¼ ρ0 e−kh

ð17Þ

is used in the trajectory design. In the research, the Gauss pseudospectral method (GPM) [40] is used to design the feasible reentry trajectory. More details about the method and the computational implementation can refer to [41]. In the simulation, the Legendre–Gauss (LG) point is set to 80. More accurate results can be obtained with more LG points at the expense of computational cost. The reentry trajectory and corresponding path constraints labeled as “normal” are presented in Fig. 4. The simulation results demonstrate that the reentry trajectories change smoothly and satisfy all the path constraints. 3.2. Closed-loop guidance by trajectory regulation Based on the obtained reentry trajectory, the guidance law can be designed. A detailed description of the design is provided in [4]. A brief review of this strategy is given here. Let x ¼ ½h ϕ θ v γ χT and u ¼ ½α sT . Define ΔxðτÞ and ΔuðτÞ as the differences between the actual and the nominal values in x and u. The linearized error dynamics of system (1)–(6) along

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Fig. 4. Reentry tajectories and corresponding path constraints.

the reference trajectory is expressed as Δ_x ¼ AðτÞΔx þ BðτÞΔu 66

ð18Þ

62

where AðτÞ∈R , BðτÞ∈R , the expression of the elements in these matrices are analytically for Eqs. (1)–(6). Their values at each τ depend on the state and control histories of the reference trajectory. Using the indirect pseudospectral method, the feedback control law Δu ¼ kðτÞΔx can easily be obtained for linear time varying system (18) (see Ref. [42]). Then, the actual reentry trajectory is controlled by guidance law: uðτÞ ¼ un ðτÞ þ ΔuðτÞ, where the asterisk denotes the reference trajectory. Note that, Δx is not generated from linearized time varying system (18) but from (1)–(6) with uðτÞ ¼ un ðτÞ þ ΔuðτÞ. 3.3. Attitude controller design based on time-varying high order sliding mode 3.3.1. Model analysis When the guidance law is determined, the design of attitude controller becomes the key to ensure the safe and stable reentry flight. In order to simplify the online calculations and the application of the sliding mode theory, the simplifications are used in the controller design [43]: since the rotational motions of the RLV are much faster than translational motions, the translational terms and the angular velocity of the Earth are neglected in attitude controller design. This simplification results in the following simplified rotational equations of motion: α_ ¼ −p cosα tanβ þ q−r sinα tanβ þ Δf 1

ð19Þ

β_ ¼ p sinα−r cosα þ Δf 2

ð20Þ

s_ ¼ −p cosα cosβ−q sinβ−r sinα cosβ þ Δf 3

ð21Þ

where Δf i ði ¼ 1; 2; 3Þ denote the uncertainties induced due to model simplification. Eqs. (7)–(9) and simplified Eqs. (19)–(21) (defined as control-oriented model) are used to design the attitude

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controller in the research. For convenience, the control-oriented model is denoted as matrix form _ ¼ −ΩðI þ ΔIÞω þ M þ ΔD ðI þ ΔIÞω

ð22Þ

_ ¼ Rω þ Δf Θ

ð23Þ

y¼Θ

ð24Þ

where ω ¼ ½p; q; r , Θ ¼ ½a; β; s , M ¼ ½M x ; M y ; M z  , Δf ¼ ½Δf 1 ; Δf 2 ; Δf 3  , ΔI and ΔD denotes the model parameters uncertainties and unknown bounded external disturbance, respectively. Eq. (24) denotes the output of the system and the matrix I;Ω;R∈R33 are defined as follows: 2 3 2 3 I xx −I xy −I xz 0 −r q 6 7 6 0 −p 7 I ¼ 4 −I yx I yy −I yz 5; Ω ¼ 4 r 5; −I zx −I zy I zz −q p 0 2 3 −cosα tanβ 1 −sinα tanβ 6 sinα 0 −cosα 7 R¼4 ð25Þ 5 −cosα cosβ −sinβ −sinα cosβ T

T

T

T

3.3.2. Time-varying high order sliding mode controller design In this section, our objective is to design controller to make the system output track the guidance commands at given convergent time t f despite uncertainties and external disturbances, equivalently satisfying the following equation: limjjα−αn jj ¼ 0;

t↦t f

limjjβ−βn jj ¼ 0;

t↦t f

limjjs−sn jj ¼ 0

ð26Þ

t↦t f

where t f is the convergent time given in advance; αn and sn denote the guidance commands, βn ¼ 0 is the desired value of sideslip angle. 3.3.2.1. Time-varying sliding mode manifold design. For brevity, the attitude control problem described by Eqs. (22)–(24) can be considered as a special case of a general MIMO nonlinear system x_ ¼ fðxÞ þ gðxÞu; n

y ¼ hðxÞ−hn ðxÞ

ð27Þ m

m

where x∈R is the measurable state vector, u∈R is the control input and y∈R is the control output vector. fðxÞ ¼ ½f 1 ðxÞ; f 2 ðxÞ; …; f n ðxÞT and gðxÞ ¼ ½g1 ðxÞ; g2 ðxÞ; …; gm ðxÞT are sufficiently smooth uncertainty functions. Now, the objective of the research is to design control input u to make y converge to zero at given convergent time t f . The relative degree vector ½r 1 ; r 2 ; …; r m T of the system (27) with respect to output vector y is assumed to be constant and known. That means the following equation can be obtained using Lie derivatives: h iT yð1r1 Þ ; yð2r2 Þ ; …; yðmrm Þ ¼ AðxÞþBðxÞu ð28Þ The rth-order sliding mode is defined through the following definition.

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Definition 1. (Levant [31]): consider the nonlinear system (27), and let the system be closed by some possible dynamical discontinuous feedback. Then, provided that yi ; y_ i ; …; yiðri −1Þ for all i ¼ 1; 2; …; m are continuous functions, and the set: 8 ðr 1 −1Þ ¼0 > < y1 ¼ y_ 1 ¼ ⋯ ¼ y1 ⋮ Γ¼ > : ðrm −1Þ ym ¼ y_ m ¼ ⋯ ¼ ym ¼0

ð29Þ

called ‘rth-order sliding mode set’, is non-empty and consists locally of Filippov’s trajectories, the motion on Γ is called ‘rth-order sliding mode’ with respect to the sliding mode variable yi . Assumption 1. (Defoort et al. [36]). The vector AðxÞ and matrix BðxÞ can be partitioned into nominal part, An ðxÞ and Bn ðxÞ, uncertain bounded part ΔAðxÞ and ΔBðxÞ. Thus, AðxÞ and BðxÞ can be expressed as AðxÞ ¼ An ðxÞþΔAðxÞ;

BðxÞ ¼ Bn ðxÞþΔBðxÞ

ð30Þ

Furthermore, matrix Bn ðxÞ is non-singular, and the uncertainty satisfies the following conditions: jjΔAðxÞ−ΔBðxÞB1 n ðxÞAn ðxÞjj≤ε1

ð31Þ

here, ε1 is known constants. Therefore, the input–output decoupling problem is solvable for system (28) by means of a control law u ¼ B−1 n ðxÞ½−An ðxÞþus 

ð32Þ

where us ¼ ½us1 ; us2 ; …; usm  denotes the discontinuous control input designed to reject the uncertainties and disturbances. Applying the control input u to Eq. (28) results in T

ðrm Þ T −1 ½y1ðr1 Þ ; y2ðr2 Þ ; …; ym  ¼ ½Im þΔBðxÞB−1 n ðxÞus −ΔBðxÞBn ðxÞAn ðxÞþΔAðxÞ

ð33Þ

Assumption 2. There is a known constant 0≤ε2 o1 such that the uncertainty in the research satisfies jjΔBðxÞB−1 n ðxÞjj≤ε2

ð34Þ

Therefore, the given time convergence of system Eqs. (22)–(24) is equivalent to the given time stabilization of the multivariable uncertain system 8 ξ_ 1i ¼ ξ2i > > > > > > ξ_ 2i ¼ ξ3i > < ⋮ ði ¼ 1; 2; …; mÞ > > _ > ξðri −1Þi ¼ ξðri Þi > > > > : ½ξ_ ðr Þ1 ; ξ_ ðr Þ2 ; …; ξ_ ðr Þm T ¼ ½Im þΔBðxÞB−1 ðxÞus −ΔBðxÞB−1 ðxÞAn ðxÞþΔAðxÞ 1

2

m

n

n

ð35Þ

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where ξðjÞi ¼ yiðj−1Þ ðj ¼ 1; 2; …; mÞ. Design the sliding manifold s ¼ ½s1 ; s2 ; …; sm T with si ð1≤i≤mÞ reading as si ðtÞ ¼ ci ξi −ci pi ðtÞ

ð36Þ pi ðtÞ ¼ ½pi ðtÞ; p_ i ðtÞ; …; pði ri −1Þ ðtÞT

with where ξi ¼ ½ξ1i ; ξ2i ; …; ξðri Þi  , ci ¼ ½c1i ; c2i ; …; cðri Þi  , cji ð1≤j≤r i Þ is positive constant. pi ðtÞ is designed to make the following hypothesis satisfied. T

T

Assumption 3. (Liu [44]).pi ðtÞ∈C ri −1 ½0; ∞Þ; p_ i ðtÞ; pï ðtÞ; …; piðri −1Þ ðtÞ∈L∞ ½0; ∞Þ, pi ðtÞ is finite in interval ½0; t f  and ξ1i ð0Þ ¼ pi ð0Þ; ξ2i ð0Þ ¼ p_ i ð0Þ; …; ξðri Þi ð0Þ ¼ pði ri −1Þ ð0Þ. Furthermore, the condition that pi ðtÞ ¼ 0; p_ i ðtÞ ¼ 0; …; piðri −1Þ ðtÞ ¼ 0 is satisfied for t≥t f . C ri −1 denotes the set of all r rank differentiable continuous functions, while L∞ ½0; ∞Þ indicates the set of all bounded functions in ½0; ∞Þ. Define a function pi ðtÞ as follows 8 ri −1 ri −1 > > k < ∑ 1ξ ðkþ1Þi ð0Þt þ ∑ pi ðtÞ k ¼ 0 k! j¼0 > > :0

ri −1

∑

! ajl

j−lþri l ¼ 0 tf

ξðlþ1Þi ð0Þ t jþri

0≤t≤t f

ð37Þ

t≥t f

where ajl ðj ¼ 0; 1; …; r i −1; l ¼ 0; 1; …; r i −1Þ is determined according to the Assumption 3. For example, in order to calculate ajl , pi ðtÞð0≤t≤t f Þ is transformed into the following form: ! t ri t ri þ1 t 2ri −1 pi ðtÞ ¼ 1 þ a00 ri þ a10 ri þ1 þ ⋯ þ aðri −1Þ0 2ri −1 ξ1i ð0Þ tf tf tf ! t t ri t ri þ1 t 2ri −1 þ a01 ri −1 þ a11 ri þ ⋯ þ aðri −1Þ1 2ri −2 ξ2i ð0Þ þ 1! tf tf tf ! t2 t ri t ri þ1 t 2ri −1 þ þ a02 ri −2 þ a12 ri −1 þ ⋯ þ aðri −1Þ2 2ri −3 ξ3i ð0Þ 2! tf tf tf ! t3 t ri t ri þ1 t 2ri −1 þ þ a03 ri −3 þ a13 ri −2 þ ⋯ þ aðri −1Þ3 2ri −4 ξ4i ð0Þ 3! tf tf tf þ⋯ þ

! t ri −1 t ri t ri þ1 t 2ri −1 þ a0ðri −1Þ þ a1ðri −1Þ 2 þ ⋯ þ aðri −1Þðri −1Þ ri ξðri Þi ð0Þ ðr i −1Þ! tf tf tf ð38Þ

i −2Þ Differentiating Eq. (38), we obtain the expressions of p_ i ðtÞ; …; pðr ðtÞ and piðri −1Þ ðtÞ. For i i −1Þ simplicity, the detail is omitted. It is obvious that the functions pi ðtÞ; p_ i ðtÞ; …; pðr ðtÞ is equal to i zero at t ¼ t f as long as the coefficients of ξ1i ð0Þ; ξ2i ð0Þ; …; ξðri Þi ð0Þ are equal to zero at t ¼ t f .

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Based on the above analysis, we obtain the following sufficient condition for pi ðtÞ ¼ 0; p_ i ðtÞ ¼ 0; …; piðri −1Þ ðtÞ ¼ 0 at given time t f : 88 a00 þ a10 þ ⋯aðri −1Þ0 þ 1 ¼ 0 > > > > > > > > > r a þ ðr i þ 1Þa10 þ ⋯ þ ð2r i −1Þaðri −1Þ0 ¼ 0 > > > > < i 00 > > > r i ðr i −1Þa00 þ ðr i þ 1Þr i a10 þ ⋯ þ ð2r i −1Þð2r i −2Þaðri −1Þ0 ¼ 0 > > > > > > > > > ⋮ > > > > > > ðri þ1Þ! ð2ri −1Þ! : > > > r i !a00 þ 2! a10 þ ⋯ þ ri ! aðri −1Þ0 ¼ 0 > > > 8 > > > a þ a11 þ ⋯aðri −1Þ1 þ 1!1 ¼ 0 > > > > > 01 > > > > > > r a þ ðr i þ 1Þa11 þ ⋯ þ ð2r i −1Þaðri −1Þ1 þ 1 ¼ 0 > > > < i 01 > > > > > > r i ðr i −1Þa01 þ ðr i þ 1Þr i a11 þ ⋯ þ ð2r i −1Þð2r i −2Þaðri −1Þ1 ¼ 0 > > > > > ⋮ > > > > > > > ð2ri −1Þ! > > : r i !a01 þ ðri þ1Þ! > 2! a11 þ ⋯ þ r i ! aðr i −1Þ1 ¼ 0 > > > 8 > > > a þ a12 þ ⋯aðri −1Þ2 þ 2!1 ¼ 0 > > <> > 02 > > > r a þ ðr i þ 1Þa12 þ ⋯ þ ð2r i −1Þaðri −1Þ2 þ 1!1 ¼ 0 > < i 02 > > > r i ðr i −1Þa02 þ ðr i þ 1Þr i a12 þ ⋯ þ ð2r i −1Þð2r i −2Þaðri −1Þ2 þ 1 ¼ 0 > > > > > > > > > ⋮ > > > > > > ðri þ1Þ! ð2ri −1Þ! > > > > : r i !a02 þ 2! a12 þ ⋯ þ ri ! aðri −1Þ2 ¼ 0 > > > > > ⋮ > > > > > > ⋮ > > > > ⋮ > > >8 > 1 > a0ðri −1Þ þ a1ðri −1Þ þ ⋯aðri −1Þðri −1Þ þ ðri −1Þ! ¼0 > > > > > > > > 1 > > > > ra þ ðr i þ 1Þa1ðri −1Þ þ ⋯ þ ð2r i −1Þaðri −1Þðri −1Þ þ ðri −2Þ! ¼0 > > > < i 0ðri −1Þ > > 1 > > > > r i ðr i −1Þa0ðri −1Þ þ ðr i þ 1Þr i a1ðri −1Þ þ ⋯ þ ð2r i −1Þð2r i −2Þaðri −1Þðri −1Þ þ ðri −3Þ! ¼ 0 > > > > > > ⋮ > > > > > > > : r i !a02 þ ðri þ1Þ! a12 þ ⋯ þ ð2ri −1Þ! aðr −1Þðr −1Þ ¼ 0 :> 2!

ri !

i

i

ð39Þ Eq. (39) consists of r i  r i algebraic equations. Solving the above equation, we can obtain a unique set of solutions. Note that the solutions of Eq. (39) can be obtained off-line and the solution is determined only by the relative degree r i . Furthermore, substituting Eq. (39) to (36), we find that si ð0Þ ¼ 0, which indicates that the system trajectories evolve on the sliding mode manifold at the initial time. Therefore, the reaching phase in sliding mode motion is eliminated and the global robustness can be guaranteed through designing a proper controller.

3.3.2.2. High order sliding mode controller design. The following theorem is proposed to ensure that the system trajectories keep on the sliding mode manifold all the time in spite of the uncertainties.

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Theorem 1. Consider the dynamic system equation (27) with a relative degree vector ½r 1 ; r 2 ; …; r m T with respect to the system output y. Given that the Assumptions 1–3 are fulfilled. Then, the control input u can be designed according to Eq. (32) with us ¼ Pf C1 M PI −GsignðsÞ

ð40Þ

with signðsÞ ¼ ½signðs1 Þ; signðs2 Þ; …; signðsm Þ . The variables Pf ; PI and CM will be given in the later. The gain G satisfies the following condition: T

G≥

ðε2 jjPf −C−1 M PI jj þ ε1 Þ þη 1−ε2

ð41Þ

with η40. This theorem guarantees that the output of system Eq. (27) tracks desired command at given convergent time t f . Proof 1. Consider a Lyapunov function candidate as V¼

1 T s s 2

ð42Þ

Taking the derivatives of V results in

  ri −1 m m m ðri Þ ðkÞ _ _ _ _ V ¼ ∑ si s_i ¼ ∑ si ðci ξi −ci p_ i ðtÞÞ ¼ ∑ si cðri Þi ðξðri Þi −pi Þ þ ∑ cki ðξki −pi Þ i¼1 i¼1 i¼1   k¼1 r i −1 m ðr i Þ ðkÞ ¼ ∑ si cðri Þi ððei þΔgi Þus þ Δf i −pi Þ þ ∑ cki ðξ_ ki −pi Þ i¼1

k¼1

ð43Þ

−1 where ei , Δgi and Δf i denote the ith  row of Im , ΔBB−1 n and ΔA−ΔBBn An , respectively. For −1 −1 simplicity, ΔBBn and ΔA−ΔBBn An are denoted by Δg and Δf respectively in the latter. Furthermore, Eq. (43) can be rewritten in the matrix form V_ ¼ sT ½CM ðus þ Δgus þ Δf−Pf Þ þ PI  ð44Þ h ðkÞ r1 −1 mÞ T _ here C M ¼ diagðcðr1 Þ1 ; cðr2 Þ2 ; …; cðrm Þm Þ, Pf ¼ ½p1ðr1 Þ ; p2ðr2 Þ ; …; pðr m  , PI ¼ ∑k ¼ 1 ck1 ðξk1 −p1 Þ; … T ; ∑rkm¼−11 ckm ðξ_ km −pðkÞ 1 Þ . Based on the Assumptions 1 and 2, we can obtain the following inequality: V_ ¼ sT ½CM ðIm þ ΔgÞus −CM Pf þ PI þ CM Δf

¼ sT ½CM ðIm þ ΔgÞðPf −C−1 M PI −GsignðsÞÞ−CM Pf þ PI þ CM Δf T ¼ s ½−CM ðIm þ ΔgÞGsignðsÞ þ CM ðI m þ ΔgÞðPf −CPI Þ−CM Pf þ PI þ CM Δf ¼ sT ½−CM ðIm þ ΔgÞGsignðsÞ þ CM ΔgðPf −C−1 M PI Þ þ CM Δf −1 ≤½−GjjCM jjjjsjj þ ε2 GjjCM jjjjsjj þ ε2 jjPf −CM PI jjjjCM jjjjsjj þ ε1 jjCM jjjjsjj ≤ð−G þ ε2 G þ ε2 jjPf −C−1 M PI jj þ ε1 ÞjjCM jjjjsjj pffiffiffiffiffiffi ≤−ηjjCM jj 2V ð45Þ since si ð0Þ ¼ 0 ði ¼ 1; 2; …; mÞ and V_ is a negative-definite function, the control law proposed in Eqs. (32) and (40) guarantees the system trajectories keep on the sliding mode manifold all the time in spite of uncertainties. Therefore, si ðtÞ ¼ ci ξi −ci pi ðtÞ ¼ ci ðξi −pi ðtÞÞ ¼ c1i ei þ c2i e_ i þ ⋯cðri Þi eiðri −1Þ ¼ 0 is always satisfied, where ei ðtÞ ¼ yi ðtÞ−pi ðtÞ. Since ei and its successive ðr i −1Þ order differential are equal to zero at the initial time, the condition ei ðtÞ ¼ e_ i ðtÞ ¼ ⋯ ¼ eri i −1 ðtÞ ¼ 0

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is always true as long as the coefficients c1i ; c2i ; …; cðri Þi are chosen such that polynomial lri þ cðri Þi lri −1 þ ⋯ þ c2i l þ c1i is Hurwitz. Since pi ðtÞ and its r i −1 order differentials are equal to zero after convergent time t f ,yi ðtÞ and its r i −1 order differentials are also equal to zero which means the rth-order sliding mode set (29) is reached at the time t f . □ 3.3.2.3. Implement issues. According to Eqs. (22)–(24), we find that the relative order of the system (22)–(24) with respect to output y is ½2; 2; 2. Thus, the relative degree of the system equals to the order of the system which means attitude control model can be linearized completely and the closed-loop system has no zero dynamics. Let Z ¼ Θ−Θn denotes the attitude tracking error, the error dynamic system is developed by repeated differentiation of Z two times as follows: n −1 _ _ _ þ Δf ΩIω−Θ̈ Þ þ RI−1 M þ ½RI−1 ðΔD−ΩΔIω−ΔIωÞ Z̈ ¼ ðRω−RI

ð46Þ

The derivative of matrix R can be calculated in view of Eq. (25) and the guidance commands Θn is generated from the guidance system [4]. It is obvious that system (46) is a special case of a general form presented in Eq. (28). Therefore, the proposed control strategy can be applied to the attitude error dynamic system (46). Furthermore, the attitude tracking can be achieved with the convergent time known in advance when the attitude controller is designed according to Eqs. (32), (40) and (41). Remark 1. It is important to note that any other feasible reentry trajectory and guidance strategy can be used in the framework of integrated guidance and control to verify the effectiveness of the proposed attitude controller.

4. Simulation results This section presents the simulation results from the integrated guidance and control developed in the previous sections to the high-fidelity six-degree-of-freedom RLV model of Section 2. Both the integrated guidance and control strategy and RLV model are written as S-function in MATLAB/Simulink. The main objective of the simulation is to demonstrate the effectiveness of the proposed control strategy in tracking the guidance commands as well as achieving safe and stable reentry flight. 4.1. Simulation parameters setting The vehicle parameters used in the integrated guidance and control are provided in Table 1. The initial reentry conditions for the 6-DOF RLV are set to be h ¼ 260; 000 ft, v ¼ 24; 061 ft/s, ϕ ¼ 01, θ ¼ 01, γ ¼ −1:0641, χ ¼ 01, α ¼ 12:601, β ¼ −11:461, s ¼ −57:291 and zero angular rate. The constant inertia matrix I and external disturbances are given as follows [47]: 2 6 I¼4

1; 997; 922

0

0

0

276; 629; 966:8

0

0

0

28; 383; 800

3 7 5;

2

1 þ sinðπt=125Þ þ sinðπt=250Þ

3

6 7 ΔD ¼ 4 1 þ sinðπt=125Þ þ sinðπt=250Þ 5  106 1 þ sinðπt=125Þ þ sinðπt=250Þ

ð47Þ

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Table 1 Vehicle parameters used in the simulation (English unit). Items

Value

Items

Value

Items

Value

Items

Value

Items

Value

μ S h0 h3

1.4076539e16 2690 1.067 −0.1903

Ω ρ0 h1 N

7.2722e−5 0.002378 −1.101 0.5

RE m h2 M

20,902,900 6309 0.6988 3.07

C CL0 CD0 CDα2

9.289e−9 −0.207 0.0785 2.04

k C Lα C Dα ΔI

4.20168e−5 1.676 −0.3529 0:1I

Fig. 5. The reentry attitude curves in Case 1.

In the simulation, the uncertainty parameter ΔI ¼ 10%I and external disturbances ΔD are added to verify the robustness of the proposed control strategy. In order to reduce the control chattering, the sign function signðsÞ in Eq. (40) is replaced by continuous saturation functions satðsÞ with boundary layers of widths as 0.01 in the simulation. 4.2. Simulation results analysis Case 1. The convergent time is set to be t f ¼ 2s. The system relative order is ½2; 2; 2. According to Eq. (39), we obtain a00 ¼ −3; a10 ¼ 2; a01 ¼ −2; a11 ¼ 1. The sliding mode parameters and control gain are set to be c1 ¼ c2 ¼ c3 ¼ ½10; 5; 1T and G ¼ 15, respectively. The simulation results labeled as ‘Case 1’ are provided in Fig. 5, illustrating that the attitude can track the guidance commands during the whole reentry flight phase. The time variations of roll, pitch and yaw angular rate and control moments are also plotted in Fig. 5. The trajectories for 6-DOF reentry vehicle are given in Fig. 4, illustrating that the actual flight trajectories can track the nominal trajectories in spite of the uncertainties and external disturbances. The heating rate, dynamic pressure and load factor curves, labeled as “Case 1” on the bottom of Fig. 4, demonstrate that the actual flight trajectories satisfy all path constraints. For better demonstrating the process of attitude tracking, the local attitude tracking curves (before 5 s) are provided in Fig. 6. From the simulation results, we find angle of attack, sideslip angle and bank angle track

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Fig. 6. The local attitude curves in Case 1.

the guidance commands at t f ¼ 2s with no overshoot. However, a large discontinuity can be observed for control moments at t f ¼ 2s. From the simulation results in Fig. 6, it can be found that the large discontinuity in control moments is caused by the rapid convergent characteristics which means the attitude tracking errors are forced to zero at the predefined convergence time t f ¼ 2s. The large discontinuity occurs in order to drive the attitude tracking error to zero in a short time. The large discontinuity in control moments means a large control effort has to be made in order to guarantee the stability of the closed-loop system. In addition, it may cause damage to actuator which is not expected in practice. To address the issue, the virtual control is introduced in Case 2.

Case 2. To avoid the drawback, the virtual control inspired by the work of Levant [30–33] is −1 ̈n introduced in this section. Define Ξ ¼ ½Ξ 1 ; Ξ 2 ; Ξ 3 T ¼ dR dt ω−RI ΩIω−Θ ; ϒ ¼ ½ϒ 1 ; ϒ 2 ; _ M ~ ¼ RI−1 M, then Eq. (46) can be rewritten as _ þ ΔfÞ, ϒ 3 T ¼ ðRI−1 ðΔD−ΩΔIω−ΔIωÞ 0 following scalar form: ~i Z ̈i ¼ Ξ i þ ϒ i þ M

ði ¼ 1; 2; 3Þ

ð48Þ

_~ M _~ T and differentiating Eq. (48) ~ ¼ ½U~ 1 U~ 2 U~ 3 T ¼ ½M ~_ 2 M Defining the virtual control as U 1 3 results in ...

Z i ¼ Ξ_ i þ ϒ_ i þ U~ i

ði ¼ 1; 2; 3Þ

ð49Þ

In this case, the system’s relative order is ½3; 3; 3. According to Eq. (39), weobtain a00 ¼ −10; a10 ¼ 15; a20 ¼ −6; a01 ¼ −6; a11 ¼ 8; a21 ¼ −3; a02 ¼ −1:5; a12 ¼ 1:5; a22 ¼ −0:5. The sliding mode parameters are chosen as c1 ¼ ½200; 50; 1T , c2 ¼ ½300; 50; 1T , ~ is c3 ¼ ½400; 50; 1T . The control gains are set to be G ¼ 20. Once the virtual control U

B. Tian et al. / Journal of the Franklin Institute 350 (2013) 1787–1807

designed, the actual control is calculated as Z ~ M ¼ IR−1 U

1803

ð50Þ

Remark 2. The missing derivatives of Z i and Ξ i ði ¼ 1; 2; 3Þ can be estimated online by means of the robust exact finite time convergent differentiator [45]. Remark 3. According to the definition of R in Eq. (25), we find that detðRÞ ¼ 1=cosβ. Since the sideslip angle β should be kept close to zero to prevent excessive heat buildup during the whole reentry phase, detðRÞ≠0 which means the matrix R is non-singular. Therefore, Eq. (50) is always ~ is designed. satisfied as long as the virtual control U The convergent time is set to t f ¼ 2s; 5s and 10s in this case. The attitude tracking curves under these conditions are plotted in Fig. 7, which illustrate that the proposed control methods can track the desired attitude commands (generated by guidance system) during the whole reentry phase. The actual flight trajectories are provided in Fig. 4, demonstrating they satisfy all the path constraints. In order to compare the dynamic response of the attitude tracking under different convergent time, the local simulation curves are given in Fig. 8. From the results, we can see that the attitude tracks the guidance commands at t f ¼ 2s; 5s and 10s, respectively. Note that the guidance commands at different convergent times are different which can be verified from the enlarged figure on the top of Fig. 8. From the body rates curves provided in Fig. 8, we find that the body rates have large magnitude, especially for the roll angular rate, when the convergence time is set to 2s. In this case, the attitude tracking errors are driven to zero within 2 s which needs a large body rate to achieve it. However, it is observed that the magnitudes of body rate decrease as the convergence time increases. From the simulation results, we can see that the amplitudes of body rates reduce to a reasonable range as the convergence time increases. In

Fig. 7. The attitude curves in Case 2.

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Fig. 8. The local attitude curves in Case 2.

addition, a notable improvement in control performance is observed when these results are compared with those provided in Fig. 7. Based on the simulation results, the main advantages of the method can be summarized as: (1) the convergent time can be set in advance which is important to achieve fast tracking for RLV; (2) the sliding mode parameters ajl ðj ¼ 0; 1; …; r i −1; l ¼ 0; 1; …; r i −1Þ are independent of the convergent time and determined only by the relative degrees of the system. It means that the convergent time is adjustable without complex calculation repeatedly as long as the relative degree of the system is determined; (3) another important information can be found that the longer the convergent time, the smaller the control amplitude, which implies the method provides an alternate solution for the flight control problem with control constrained. Although this problem is not discussed in the research, it may be feasible to adjust the convergent time to adapt to different control constraints. Remark 4. With respect to the choice of convergent time t f , there is theoretically no limitation provided that 0ot f o∞. However, by a practical point of view, the choice of t f is related to the physical limits of the control and the dynamics of the system. Remark 5. The attitude tracking can be achieved in a finite time t f as long as the condition G≥ðε2 ∥Pf −C−1 M PI ∥ þ ε1 Þ=ð1−ε2 Þ þ η is hold during the interval t∈½0; t f . However, the finite time tracking with convergence time known cannot be achieved if the upper bound of uncertainty and external disturbance are not exactly known in advance. Moreover, the traditional adaptive

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_^ ¼ ∥s∥=p provided in Ref. [46] cannot be applied to the proposed control strategy method G −1 ^ directly. The condition G≥ðε 2 ∥Pf −CM PI ∥ þ ε1 Þ=ð1−ε2 Þ þ η will not be satisfied at the beginning if the adaptive method is used. In this case, the attitude tracking cannot be achieved at predefined convergence time. 5. Conclusions and future works In this paper, the nonlinear robust controller is developed for a reusable launch vehicle that achieves safe and stable reentry flight despite having model uncertainties and external disturbances. The global robustness of the attitude control system is ensured via designing timevarying sliding mode manifold which makes the system trajectory evolve on the manifold at the initial time. Then, the controller is designed to assure robust tracking of the guidance commands at given convergent time for attitude control system. Furthermore, in order to reduce the control saltation, the virtual control is introduced to design the attitude controller for RLV. 6-DOF simulation results are presented to demonstrate the effectiveness of the proposed integrated guidance and control strategy in tracking the guidance commands as well as achieving safe and stable reentry flight. In the future, the research will focus on the transformation of control matrix to control surface deflections and the attitude control for RLV in fault mode. Acknowledgments This work has been supported by National Natural Science Foundation of China (61203012, 61273092, 91016018), Key Grant Project of Chinese Ministry of Education (No. 311012), Tianjin Research Program of Application Foundation and Advanced Technology (11JCZDJC25100, 12JCZDJC30300), Aeronautical Science Foundation of China (20125848004) Supported by Science and Technology on Aircraft Control Laboratory.

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