IAA-AAS-DyCoSS-4 -- A GLOBAL SLIDING MODE CONTROL WITH PRE-DETERMINED CONVERGENCE TIME DESIGN FOR REUSABLE LAUNCH VEHICLES IN REENTRY PHASE L. Wang, Y. Z. Sheng, X. D. Liu, and P. L. Lu INTRODUCTION This paper addresses the robust attitude control problem of reusable launch vehicles (RLV) with parametric uncertainties and external disturbances in reentry phase. Based on the feedback-linearized RLV rotational equations of motion, a global sliding mode control (SMC) strategy with a priori stated convergence time is first proposed, which ensures the global robustness of the controlled system. As compared with the existing sliding mode attitude control method, an improved system performance can be obtained in terms of convergence time and robustness. Then, the steady-state error problem caused by the continuous approximation techniques such as the boundary layer method (which is used to alleviate the control chattering) is considered. A disturbance observer (DO) based global SMC strategy is presented to improve the control accuracy. Finally, the validity of the proposed strategies is verified by both theoretical analysis and simulation results for the attitude control problem of X-33 RLV in reentry phase. Reusable launch vehicles (RLV) are designed to perform multiple missions and to be reusable, which will reduce the cost of access to space dramatically. The whole reentry flight is generally separated into three primary phases: Reentry phase (from point A to B), Terminal Area Energy Management (TAEM) phase (from point B to C), and Approaching and Landing phase (from point C to ground). An illustration of the reentry flight trajectory for a RLV is shown in Figure. The flight control of RLV in reentry phase has attracted considerable interest for its critical position in accomplishing the reentry tasks safely. However, the reentry flight control involves attitude maneuvering through a wide range of flight envelope, and suffers from various kinds of external disturbances and modeling uncertainties. These, together with the facts that RLV are characterized by its poorly understood aerodynamic qualities, high nonlinearity and coupling characteristics in both attitude dynamics and kinematics, increase the complexity of the attitude control design. Therefore, in order to achieve accurate and rapid attitude tracking, nonlinear robust control techniques are more suitable than linear control methods, e.g., gain scheduling. Ph.D candidate, School of Automation, Beijing Institute of Technology, and School of Automation, 3 staff, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, 8, P. R. China. Email: nicholas33@bit.edu.cn Lecturer, School of Automation, Beijing Institute of Technology, and School of Automation, 3 staff, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, 8, P. R. China. Lecturer, School of Automation, Beijing Institute of Technology, and School of Automation, 3 staff, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, 8, P. R. China. Professor, School of Automation, Beijing Institute of Technology, and School of Automation, 3 staff, Beijing Institute of Technology, 5 South Zhongguancun Street, Haidian District, Beijing, 8, P. R. China.
Figure. Example reentry flight trajectory for a RLV Sliding mode control (SMC), as one of effective robust control algorithms, has been widely used in the reentry attitude controller design due to its attractive features such as easy implementation, insensitivity to parametric uncertainty and external disturbance satisfying the matching condition, and fast dynamic response., 3, 4 In References, 3 and 4, time-scale separation principle and conventional SMC strategy (i.e., the phase plane trajectory of controlled system consists of two parts: the reaching phase and the sliding phase) are combined to develop the attitude controller. Due to the u- tilization of the conventional SMC strategy, the resultant controller can only ensure the asymptotical convergence of tracking errors. The controlled system is still sensitive to the matched lumped uncertainties (including parametric uncertainties and external disturbances) during the reaching phase. Moreover, as reported in Reference 5, the adoption of time-scale separation introduces modeling errors because of the original coupling characteristic of the rotational dynamics and kinematics. In this paper, the robust attitude control problem of RLV in the reentry phase is addressed. As in Reference 5, feedback linearization (FBL) approach is first adopted to handle the inherent nonlinearity and coupling lying in the rotational dynamics and kinematics, by which the use of time-scale separation principle is avoided. Then, by appropriately designing the sliding surface, a global SMC strategy with pre-determined convergence time is developed based on the the feedback-linearized reentry attitude control problem, which eliminates the reaching phase and achieves the global robustness. As a result, the proposed control is superior to the preceding SMC based attitude controller, 3, 4 in terms of convergence time and robustness. Furthermore, the global chattering problem existing in the proposed algorithm is considered. To reduce the steady-state error caused by the continuous approximation techniques such as the boundary layer method, a composite scheme, namely disturbance observer (DO) based global SMC algorithm, is further presented. By incorporating the disturbance observer into the related control design, a more precise tracking performance will be achieved. The effectiveness of the proposed attitude control schemes is verified through the application of them to the attitude control problem of X-33 RLV in reentry phase.
PRELIMINARIES Mathematical Model This paper considers the attitude control problem of an unpowered, lifting-body atmospheric RLV in reentry phase. The related rotational equations of motion (i.e., attitude dynamics and kinematics) are described as: 5, 6 I ω + ω Iω = M + M d () Ω = R(Ω)ω + f () where I is the symmetric positive definite inertia matrix and is defined as I xx I xz I = I yy I xz I zz ω = [p, q, r] T is the angular rate vector with p, q, r being the roll, pitch, and yaw rates respectively, the superscript ( ) denotes the skew-symmetric matrix operator on any vector a = [a, a, a 3 ] T such that a 3 a a = a 3 a a a M = [M x, M y, M z ] T is the vector of control moment with M x, M y and M z being the roll, pitch, and yaw moments respectively, M d R 3 denotes the vector of bounded external disturbance moment. Ω = [α, β, µ] T is the aerodynamic angle vector defined as angle of attack, sideslip angle, and bank angle respectively, R(Ω) R 3 3 is defined by: cos α tan β sin α tan β R(Ω) = sin α cos α cos α cos β sin β sin α cos β f = [ f, f, f 3 ] T denotes unknown but bounded smooth vector-function, which is mainly caused by the model simplification procedure. 6 Moreover, the uncertainty on the inertia matrix is taken into account, i.e., I = I + I with I being the nominal inertia matrix and I being the unknown but bounded inertia variation. Based on the above discussion, Eq. () can be rewritten as I ω = ω I ω + M + d (3) where d = I ω ω Iω + M d. It is assumed that d is bounded. For clarity, related arguments of various functions are be ignored in the following content. Feedback Linearization Choosing the control moments M x, M y, M z and the aerodynamics angles α, β, µ as the control inputs and the outputs respectively, the attitude kinematics and dynamics described by Eqs. () and (3) can be expressed as: ẋ = f(x) + 3 g i (x)u i + d i= (4) y i = h i (x), i =,, 3. 3
where x = [p, q, r, α, β, µ] T is the state vector, M = [u, u, u 3 ] T = [M x, M y, M z ] T and y = [y, y, y 3 ] T = [h, h, h 3 ] T = [α, β, µ] T denote the vectors of control input and output respectively, and f (x) f (x) f(x) = f 3 (x) f 4 (x) = f 5 (x) f 6 (x) (I xx I yy +I zz )I xz I pq + (I yy I zz )I zz I xz I qr I xz I yy (r p ) + I zz I xx I yy pr pq + ( I xx+i yy I zz )I xz I qr, p cos α tan β + q r sin α tan β p sin α r cos α p cos α cos β q sin β r sin α cos β (I xx I yy )I xx +Ixz I g (x) = [ I zz I,, I xz I,,, ]T, g (x) = [,,,,, ] T, g 3 (x) = [ I xz I yy I,, I xx,,, ]T I with I = I xx I zz I xz. d = [(I d)t, f T ] T R 6 denotes the system uncertain term. When the FBL technique is applied to Eq. (4), each output y i needs to be differentiated a sufficient number of times until a control input u i appears in the resulting equation. After differentiating each output y i twice, the output dynamics for y is described by ÿ L f ÿ h L g L f h L g L f h L g3 L f h = L f h + L g L f h L g L f h L g3 L f h M + v ÿ 3 L f h 3 L g L f h 3 L g L f h 3 L g3 L f h 3 }{{}}{{} F E =F + EM + v where v = [ v, v, v 3 ] T = RI d + f stands for the lumped uncertainty resulting from the FBL procedure. As the external disturbance, inertia matrix variation and system states are all bounded, it is assumed that v is bounded by an upper bound, i.e., there exists a constant v max R +, such that v < v max with being the vector infinity norm. In view of Eq. (5), the vector relative degree of system (4) is [,, ], and hence the total relative degree (i.e., + + = 6) equals to the order of the system. Moreover, since the sideslip angle β is kept close to during reentry, det(e) = cos β sin β tan β I I yy I I yy I I yy. Therefore, system (4) can be linearized completely without zero dynamics. 7 Define the feedback control law as M = E ( F + v) (6) with v = [v, v, v 3 ] T being the auxiliary control input. Then, by substituting Eq. (6) into Eq. (5), the output dynamics of system (4) is transformed into the following decoupled integrator form: (5) ÿ = v + v (7) It should be noted that Eq. (7) can be viewed as three decoupled second-order system and each output y i, i =,, 3, is independently controlled by the corresponding input v i. 4
Problem Statement In this paper, the general control design goal is to determine the control moment vector M such that the commanded aerodynamic angle profiles y c = [α c, β c, µ c ] T are robustly tracked with predetermined convergence time in the presence of bounded parametric uncertainties (i.e., inertia variations I), model simplification errors f, and external disturbance torques M d. In the sequel, the proposed global SMC laws are firstly designed in terms of the auxiliary control v, then the control moment M is obtained through Eq. (6). Specifically, the attitude control problem during reentry is to determine the control input v in the global SMC framework for the resultant FBL model in Eq. (7), such that the output vector y tracks the commanded aerodynamic angle profiles y c in finite time. That is, lim e = lim (y y c ) = (8) t t f t tf where e = y y c = [e, e, e 3 ] T is the vector of tracking error, and t f is the pre-determined convergence time. The functional diagram of the proposed global SMC-FBL attitude control design is presented in Figure. Parametric Uncertainties/External Disturbances Attitude Controller Commanded Aerodynamic Angle Profiles Global SMC Strategy FBL Flight States RLV model Figure. Diagram of the global SMC-FBL attitude control design MAIN RESULTS Global SMC Design Based on the FBL model in Eq. (7), a global SMC strategy with pre-determined convergence time is developed in this subsection. Firstly, a novel sliding function S = [s, s, s 3 ] T is designed as S = (ė P (t)) + K(e P (t)) (9) where K = diag(k, k, k 3 ) with its element k i, i =,, 3, being positive constant, P (t) = diag(p (t), p (t), p 3 (t)) with p i (t), i =,, 3, satisfying the following assumption in order to ensure the finite time convergence of the tracking errors. Assumption. The function p i (t), i =,, 3, is finite in interval [, t f ] and satisfies: ) p i (t) C [, ], ṗ i (t), p i (t) L, where C [, ) denotes the set of all rank differentiable continuous functions defined in [, ), and L indicates that the set of all bounded functions in [, ); 5
) p i () = e i () and ṗ i () = ė i () to ensure s i () = ; 3) p i (t) = ṗ i (t) = p i (t) = when t t f. Then, a control law that makes the sliding surface in Eq. (9) attractive should be designed. To this end, the control law is given by v = ÿ c + P (t) K(ė P (t)) ηsgn(s) () where η = diag(η, η, η 3 ) is the switching gain matrix with its elements η i depending on v max, e.g., η i = v max + δ with δ being a small positive constant, sgn( ) is the vector sign function and sgn(s) = [sgn(s ), sgn(s ), sgn(s 3 )] T. Theorem. Suppose that the Assumption is satisfied. Then, for the feedback-linearized attitude control problem of RLV described in Eq. (7), by adopting the sliding surface function defined in Eq. (9) and the corresponding control law in Eq. (), a global sliding mode is achieved. Moreover, the attitude tracking error e is driven to zero at the pre-determined convergence time t f. Proof. Consider the Lyapunov function candidate By taking the time derivative of V, one gets V = ST S () V =S T Ṡ =S T [(ë P (t)) + K(ė P )] =S T [v + v ÿ c P (t) + K(ė P )] () Substituting the control law () into Eq. () yields V =S T ( ηsgn(s) + v) 3 ( η i s i + v s i ) i= 3 ( η i s i + v max s i ) i= δ 3 s i i= (3) Clearly, for any S(t) R 3, V is non-positive and hence V (t) V (). In view of the definition of the sliding surface in Eq. (9) and the Assumption, the initial value of S is zero (i.e., S() = ), which result in V () =. Then, one gets V. On the other hand, it is obvious that V for any S(t) R 3 according to Eq. (). Consequently, it is easy to conclude that V, which implies that S(t), t [, ). That is, a global sliding mode is achieved. Based on the above discussion, S(t) = (ė P (t)) + K(e P (t)) is satisfied for all t [, ). Since the initial values of e P (t) and ė P (t) are both equal to zero according to Assumption, it can be easily concluded that e P (t) = ė P (t) = for all t [, ) provided that the elements of K are positive constants. Therefore, the tracking error e and its derivative ė will both be driven to zero at the previously stated time t f as p i (t) = ṗ i (t) = for t t f. 6
Remark. It should be pointed out that the proposed sliding surface in Eq. (9) is similar to that in Reference 8. However, there exists large discontinuity (jump) in the control moments when the control algorithm in Reference 8 is used. To avoid this problem, virtual control is introduced in Reference 8, and consequently robust exact finite time convergent differentiator 9 has to be incorporated into the related control design to estimate the missing derivatives of system states. In this paper, the control discontinuity problem is solved by simply adding the condition p i (t t f ) = as shown in the Assumption, which is simpler and more effective than the way used in Reference 8. Remark. With respect to the selection of the function p i (t), i =,, 3, any functions that satisfy Assumption can be adopted. One possible choice of p i (t) is given by: { a 4i t 4 + a 3i t 3 + a i t + a i t + a i t < t f p i (t) =, (4) t t f whose parameters a i, a i, a i, a 3i and a 4i can be determined according to Assumption. Specifically, a i = e i (), a i = ė i (), a i = 3ė i()t f 6e i () t, f DO based Global SMC Design a 3i = 3ė i()t f + 8e i () t 3 f, a 4i = ė i()t f 3e i () t 4. f In real implementation, the discontinuous control laws given in Eq. () may result in control chattering due to imperfections in switching devices. Moreover, due to the global sliding phase feature of the proposed control strategy, the chattering phenomenon exists from the very beginning of the control action. To alleviate the chattering phenomenon, boundary layer technique is used to give a continuous approximation of the discontinuous sign function sgn( ). That is, replace the sign function sgn( ) in Eq. () by saturation function sat(s) = [sat(s ), sat(s ), sat(s 3 )] T, which is defined as follows: sat(s i ) = { si κ i if s i κ i, i =,, 3, (5) sgn(s i ) if s i > κ i where κ i R + denotes the boundary layer thickness. In such a case, the tracking error e will not converge to zero in finite time but rather can only be expected to converge into some small vicinity of the origin depending on the boundary layer thickness κ i in finite time. As a result, the chattering is reduced at the cost of degraded control accuracy and robustness. In order to enhance the control accuracy and system robustness, a DO based global SMC strategy is developed in this subsection, whose structure is illustrated in Figure 3. z is the estimate of the lumped uncertainty v and will be determined below. In the sequel, the following assumption is needed. Assumption. The variables S, v, ÿ c and ė are assumed to be bounded and Lebesgue-measurable respectively, the lumped uncertainty v is times differentiable and bounded, i.e., there exists a constant l i, i =,, 3 such that v i l i. Based on Assumption, the structure of the adopted DO is given in the following lemma. 7
Parametric Uncertainties/External Disturbances Attitude Controller Commanded Aerodynamic Angle Profiles DO based Global SMC Strategy FBL Flight States DO RLV model Figure 3. Structure of the DO based global SMC strategy Lemma. 9,, Consider the following DO: ż =χ + v ÿ c p(t) + K(ė ṗ(t)), z s /3 sgn(z s ) χ = γ L /3 z s /3 sgn(z s ) + z, z 3 s 3 /3 sgn(z 3 s 3 ) ż =χ, z χ / sgn(z χ ) χ = γ L / z χ / sgn(z χ ) + z, z 3 χ 3 / sgn(z 3 χ 3 ) sgn(z χ ) ż = γ 3 L sgn(z χ ), sgn(z 3 χ 3 ) (6) where z = [z, z, z 3 ] T, z = [z, z, z 3 ] T, z = [z, z, z 3 ] T, χ = [χ, χ, χ 3 ] T, χ = [χ, χ, χ 3 ] T, γ, γ, γ 3 and L = diag(l, l, l 3 ) are the design parameters of the observer (7). Supposing that Assumption is satisfied and the related parameters l i, γ i, i =,, 3, are chosen properly (e.g., γ i can be chosen recursively as suggested in Reference 9), then in the absence of measurement noise the following equalities are established in finite time: z = S, z = v, z = Proof. The proof is omitted, which is similar to References and. v. (7) According to structure of the DO based global SMC strategy shown in Figure 3, the DO based global SMC algorithm is designed as v = ÿ c + P (t) K(ė P (t)) ηsgn(s) z (8) Where η is selected the same as that in Eq. (), and z is obtained from Eq. (6). 8
The closed-loop stability under the DO based global SMC algorithm (8) can be easily proved as the DO is finite-time convergent. Therefore, the proofs is omitted here and next we will focus on the study of the behavior inside the boundary layer of both the global SMC and DO based global SMC algorithms to show the improvement of the DO based global algorithm in control accuracy. For the DO based global SMC algorithm, the time derivative of the sliding function in Eq. (9) is given by Ṡ = ηκ S + v z (9) where κ = diag(κ, κ, κ 3 ). On the other hand, the time derivative of the sliding function (9) controlled by the global SMC algorithm is described as Ṡ = ηκ S + v () Denote the static value of S controlled by the DO based global SMC and the global SMC algorithms as S d and S o, respectively. Then, by letting Ṡ = in Eqs. (9) and (), one gets { S d = η κ( v z ) S o = η κ v () From Eq. (), it is obvious that the static value of S controlled by the DO based global SMC algorithm is much smaller than the global SMC algorithm after the convergence of the DO. In view of the definition of S, small value of S implies a more precise tracking accuracy is achieved. Consequently, an improvement in control accuracy can be provided by the DO based global SMC algorithm. SIMULATION RESULTS Reference Reentry Trajectory Design To achieve the reentry flight of RLV, the Radau Pseudospectral Method (RPM) is used to generate a feasible reference reentry trajectory, in which the used equations of the motion are given by ḣ =V sin γ V cos γ sin χ ϕ = (h + R e ) cos χ V θ = cos γ cos χ (h + R e ) V = D m g sin γ, () γ = mv [L cos µ m(g V ) cos γ] (h + R e ) χ = L sin µ mv cos γ + V cos γ sin χ tan θ (h + R e ) where h is the altitude, ϕ is the latitude, θ is the longitude, V is the velocity, γ is the flight path angle, χ is the heading angle, R e is the radius of earth, g = µ /(h + R e ) is the inverse-square 9
gravitational acceleration with µ =.476539 7 (ft 3 /s ), L and D denote the lift and drag forces, respectively, which are given by L = ˆqS ref C L (α), D = ˆqS ref C D (α), where S ref denotes the reference area, and ˆq = ρv denotes the dynamic pressure with ρ being the atmospheric density. The detailed expressions of ρ, C L (α) and C D (α) can be found in Reference 3. When the reentry trajectory design target is set to maximize the cross range, the objective of trajectory design is then to find a control history u = [α, µ] T that minimizes the following cost function: J = θ f. (3) subject to the dynamic model given by Eq.(). For achieving this aim, the initial and final conditions are specified as: (h, ϕ, θ, V, γ, χ ) = (6 ft, deg, deg, 56 (ft/s), deg, 9 deg), (h f, V f, γ f ) = (8 ft, 5 (ft/s), 5 deg). (4) where the subscripts and f are the initial and final time instant, respectively. Moreover, the path constraints associated with heating-rate Q, dynamic pressure ˆq, and structural load N are given by: Q Cρ n V n (l + l α + l α + l 3 α 3 ) Btu/ft /s ˆq = ρv 8 lb/ft (5) N L + D /(mg).5 where n =.5, n = 3.7, C = 9.89 9 Btu s.7 /ft 3.57 /slug.5 and l =.67, l =., l =.6988, l 3 =.93. Figure 4 shows the generated reentry trajectories, corresponding path constraints and aerodynamic angle profiles. It can be seen that the associated path constraints (i.e., heating-rate Q, dynamic pressure ˆq, and structural load N) are all satisfied. Besides, the generated reentry trajectories and the control variables such as angle of attack α and bank angle µ, all have a relatively smooth change. During the reentry phase, the sideslip angle β is kept to zero to prevent the excessive heat buildup. Attitude Control Simulation Results As this paper mainly focuses on the attitude control of RLV in the reentry phase, the aerodynamic angle profiles generated in the preceding subsection is directly used here as the commanded aerodynamic angle profiles to be tracked. In this subsection, numerical simulations are conducted to compare the effectiveness of the proposed strategies for RLV in the presence of parametric uncertainties and external disturbances. The global SMC algorithm in Eq. () and the DO based global SMC algorithm in Eqs. (8) and (6) are applied to the attitude control problem under study for comparison and referred as GSMC, and DO-GSMC for clarity, respectively. In the simulation, the 6-DOF mathematical model given in Reference 4 is used. The initial system states are: the altitude h() = 6 ft (i.e., the initial radial position is h() + R e ),
3 x 5 4 h (ft) φ (deg) 5 θ (deg) 3 x 4 5 V (ft/s) γ (deg) 5 χ (deg) 5 Q (Btu/ft /s) α (deg) 5 5 8 6 ˆq (lb/ft ) β (deg) 5 5.5.5 N µ (deg).5.5 4 6 8 Figure 4. System behavior of the simulated RLV. The first two rows are for the generated reentry trajectories, the third row is for the corresponding path constraints, and the last row is for the aerodynamic angle profiles
the latitude ϕ() = deg, the longitude θ() = deg, the velocity V () = 56 (ft/s), the flight path angle γ() = deg, the heading angle χ() = 9 deg, the aerodynamic angles Ω() = [,, 65] T deg, the angular rate ω() = [,, ] T (deg/s). The nominal inertia matrix of the X-33 RLV during the reentry phase is given by 5 4347 788 I = 96 (slug ft ), 788 354 and the uncertainty is introduced into the inertia matrix by about % of the nominal value. Besides, the form of external disturbance considered in the simulation is given by + sin(πt/5) + sin(πt/5) M d = + sin(πt/5) + sin(πt/5) 5 (lb ft), + sin(πt/5) + sin(πt/5) which is added to each axis directly. Table shows the related control parameters. Table. Related control parameters Controllers Parameters GSMC DO-GSMC Sliding function gain: K diag(,, ) diag(,, ) Pre-determined convergence time: t f (s) Control parameter: η diag(,, 3) diag(,, 3) Control parameter: κ i, i =,, 3.,.,..,.,. DO gain: γ i, i =,, 3 Null 8.4, 4.,. DO gain: L Null diag(.3,.3,.) The simulation results are shown in Figures 5 to 8. Figures 5 and 6 depict the time histories of aerodynamic angles tracking, angular rates and control moments controlled by the GSMC and DO-GSMC algorithms respectively, from which it can be seen that both of the controllers have successfully accomplished the attitude tracking. A detailed comparison of the tracking error responses controlled by those two algorithms are shown in Figure 7, where the local curves of the tracking errors are also provided for a better comparison. It can be seen that the tracking errors are all convergence to zero at t = s, which is consistent with the previously stated convergence time t f. Besides, the tracking accuracy controlled by the DO-GSMC algorithm is higher than that controlled by the GSMC algorithm, which verifies the precision improvement of the DO-GSMC algorithm. Figure 8 demonstrates the comparison of the sliding surface responses controlled by those two algorithms, form which it can also be seen that the magnitudes of S in the DO-GSMC is smaller than that of the GSMC algorithm, even the same boundary layer thicknesses are used. CONCLUSION In this paper, a global SMC strategy with a priori stated convergence time has been proposed to achieve the robust attitude control of RLV in the presence of parametric uncertainties and external disturbances in reentry phase. Moreover, in order to address the steady-state error problem caused by the continuous approximation techniques such as the boundary layer method, a DO is
5 x 3 α (deg) 5 β (deg) 5 µ (deg) 4 α c α 5 β c β 6 µ c µ p (deg/s) Mx (lb.ft) 8 6 4 x 5 q (deg/s) My (lb.ft) 6 4 3 x 5 r (deg/s) Mz (lb.ft).5.5 x 4 Figure 5. The system responses controlled by the GSMC algorithm. The first row represents the time histories of aerodynamic angles tracking, the second row corresponds to the angular rate responses, and the third row is for the control moments. 3
5 x 3 α (deg) 5 β (deg) 4 µ (deg) 4 α c α 5 6 β c β 8 6 µ c µ p (deg/s) Mx (lb.ft) 8 6 4 x 5 My (lb.ft) q (deg/s) 6 4 x 5 3 r (deg/s) Mz (lb.ft).5.5 x 4 Figure 6. The system responses controlled by the DO-GSMC algorithm. The first row represents the time histories of aerodynamic angles tracking, the second row corresponds to the angular rate responses, and the third row is for the control moments. 4
e (deg) e (deg) 4 4 GSMC 6 DO GSMC 8 local curve 4 4 6 8 3 4 5 local curve..5 e (deg) e (deg) 5 x 3 local curve.... 3 4 5 local curve..5 e3 (deg) e3 (deg) 5 5 local curve 5 5 3 4 5 local curve..5 e (deg).5 e (deg).5 e3 (deg).5... 5 5 5 Figure 7. Comparison of tracking errors, where the first row corresponds to the global curves, the second and third rows represents the local curves. 5
5 x 3 GSMC DO GSMC s 5 5 5 5.5 x 3 s.5 5 5 5 5 x 3 s3 5 5 5 5 Figure 8. Comparison of sliding surface responses. further incorporated into the proposed global SMC strategy to improve the tracking accuracy. The effectiveness of the proposed strategies has been verified from both the theoretical analysis and simulation results. ACKNOWLEDGMENT The authors acknowledge the financial support from Major State Basic Research Development Program [grant number CB7], National Natural Science Foundation of China [grant number 3734], and Innovative Research Team of Beijing Institute of Technology. REFERENCES [] S. D. Ridder and E. Mooj, Terminal area trajectory planning using the energy-tube concept for reusable launch vehicles, Acta Astronautica, Vol. 68,, pp. 95 93. [] Y. Shtessel, J. McDuffie, M. Jackson, C. Hall, M. Gallaher, D. Krupp, and N. D.Hendrix, Sliding mode control of the X-33 vehicle in launch and re-entry modes, Proceedings of AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA 998-444. [3] Y. Shtessel, C. Tournes, and D. Krupp, Reusable launch vehicle control in sliding modes, Proceedings of AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA 997-3533. [4] Y. Shtessel, C. Hall, and M. Jackson, Reusable launch vehicle control in multiple-time-scale slding modes, Journal of Guidance, Control, and Dynamics, Vol. 3, No. 6,, pp. 3. [5] W. R. V. Soest, Q. P. Chu, and J. A. Mulder, Combined feedback linearization and constrained model predictive control for entry, Journal of Guidance, Control, and Dynamics, Vol. 9, No., 6, pp. 47 434. [6] J. J. Recasens, Q. P. Chu, and J. A. Mulder, Robust model predictive control of a feedback linearization system for a lifting-body re-entry vehicle, Proceedings of AIAA Guidance, Navigation and Control Conference and Exhibit, AIAA 5-647. [7] Nonlinear control systems. Berlin: Springer-Verlag, 3nd ed., 995. 6
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