- Email: [email protected]
Launch vehicle attitude control using sliding mode control and observation techniques
Available online at www.sciencedirect.com
Journal of the Franklin Institute 349 (2012) 397–412 www.elsevier.com/locate/jfranklin
Launch vehicle attitude control using sliding mode control and observation techniques James E. Stotta, Yuri B. Shtesselb,n a
Safety and Mission Assurance Directorate, QD33, NASA, Marshall Space Flight Center, AL 35812, USA b Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 2 March 2011; received in revised form 6 June 2011; accepted 28 July 2011 Available online 16 August 2011
Abstract In determining flight controls for launch vehicle systems, several uncertain factors must be taken into account, including a variety of payloads, a wide range of flight conditions and different mission profiles, wind disturbances and plant uncertainties. Crewed vehicles must adhere to human rating requirements, which limit the angular rates. Sliding mode control algorithms that are inherently robust to external disturbances and plant uncertainties are very good candidates for improving the robustness and accuracy of the flight control systems. Recently emerging Higher Order Sliding Mode (HOSM) control is even more powerful than the classical Sliding Mode Controls (SMC), including the capability to handle systems with arbitrary relative degree. This paper proposes sliding mode launch vehicle flight controls using classical SMC driven by the sliding mode disturbance observer (SMDO) and higherorder multiple and single loop designs. A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is shown. The intensive simulations demonstrate efficacy of the proposed HOSM and SMC-SMDO control algorithms for launch vehicle attitude control. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Launch Vehicles, in general terms, are vehicles with the primary purpose of carrying a payload from the surface of the Earth into outer space. Outer space, in contrast to air space, is most commonly defined as everything beyond the Karman line 100 km n
Corresponding author. Tel.:þ12568246164; fax:þ12568246803. E-mail addresses: [email protected] (J.E. Stott), [email protected] (Y.B. Shtessel).
0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.07.020
398
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
Nomenclature q ¼ fq0 ,q1 ,q2 ,q3 gT quaternion vector x ¼ fp,q,rgT body rate vector (rad/s) J matrix of inertia sðÞ ,sðÞ sliding variables T control torque (moment) (lbs-ft) Fð Þ the bounded disturbance that contains the disturbance torque, launch vehicle inertia variations, and aerodynamic surface and engine failures
(62 miles) above the Earth’s surface. This line roughly marks the altitude at which the density of the atmosphere of the Earth is so low that a vehicle would have to fly faster than orbital speed, i.e., the speed tangential to the Earth’s surface, which is required to maintain an orbit of the Earth at that altitude, in order to support flight using aerodynamic lift. Launch Vehicles can support flight at altitudes above the Karman line since they use thrust instead of aerodynamic lift to overcome the force of gravity. Launch Vehicles are generally classified as either Expendable or Reusable (or partially reusable), orbital or sub-orbital, and manned or unmanned. Expendable Launch Vehicles (ELVs) are designed to be for one-time use and normally fall back to Earth once the payload is deployed breaking apart in the Earth’s atmosphere. Reusable Launch Vehicles (RLVs), or reusable components of Partially Reusable Launch Vehicles (P-RLVs), are designed to perform many missions within their life cycle. Orbital Launch Vehicles can carry a payload into a stable orbit about the Earth, whereas a suborbital Launch Vehicle cannot. Other characterizations of Launch Vehicles are mass to orbit ratios, number of stages, and type of propulsion (e.g., solid or liquid). Examples of several types of launch vehicles are shown in Fig. 1. As Launch Vehicles continue to evolve along with the development of new technologies, the main objectives of new designs are to increase performance, reliability, safety, and availability, while decreasing cost. Technologies such as structures, avionics, health management systems, and ground operations are studied to determine optimal ways of reducing cost and risks associated with Launch Vehicle missions. As computational capability increases, control algorithms play an ever more important role as higher levels of performance, reliability, safety, and availability are achieved. Currently, the most ubiquitous control algorithm used is the Proportional Integral Derivative (PID) control with gain scheduling. PID control was used on the Saturn V and the Space Shuttle, and is being developed in the Ares rockets. Although many updates have taken place over the last 30 years, PID with gain scheduling control, remains dominant due to its proven heritage. In the late 1990s, NASA’s Marshall Space Flight Center (MSFC) led an Advanced Guidance and Control (AG&C) project funded by the X-33 Program Office to study ascent and entry controllers. This work continued under the Integration and Testing of Advanced Guidance and Control Technologies (ITAGCT) for the Space Launch Initiative (SLI) Program. The goal of the flight control system concepts that were developed under NASA’s AG&C project was to seek new launch vehicle control technologies to successfully fly in an expanded range of flight conditions, vehicle parameter variations, mission profiles, and unplanned failures requiring in-flight adaptation, as well
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
399
Fig. 1. Launch vehicles.
as to significantly reduce the number of Guidance, Navigation, and Control (GN&C) tuning parameters [1]. Several control techniques, including Proportional-Integral-Derivative (PID) control law with gain scheduling [2,3], Trajectory Linearization Control (TLC) [4,5], adaptive Neural Network (NN) [6,7], and Theta-D [22] launch vehicle control algorithms have been developed under AG&C and used for launch vehicle flight control. However, the robustness issues are still not fully addressed by means of these control techniques. The Sliding Mode Control (SMC) algorithm [8,9,24–28], is a Lyapunov function based technique, providing global stability thus avoiding gain scheduling. It is an attractive design because of its inherent insensitivity and robustness to unknown plant uncertainties and external disturbances. Furthermore, its robustness accommodates different trajectories and aerodynamic surface and engine failures without gain scheduling. Due to these advantages over the PID and other controllers, the SMC is believed to reduce risk and dramatically decrease the amount of time spent in pre-flight analysis, thus reducing cost over current values [10,11,16,17]. Recently emerging Higher Order Sliding Mode (HOSM) control [12–15,23] is even more powerful than the classical SMC, including the capability to handle systems with arbitrary relative degree that allows avoiding multiple loop control system design with partial or full dynamical collapse (i.e. reduction of the order of the compensated dynamics). By increasing the system’s relative degree artificially, sliding control of arbitrarily smooth order can be achieved, which completely removes the chattering effect. This HOSM approach allows achieving control smoothness without approximation of sign function by a saturation one (as it often done in classical SMC), while retaining the main properties of the classical SMC and improving its accuracy. HOSM works with the discontinuous control acting on the higher-order time derivatives of the sliding variable, instead of working on the first time derivative as happens in classical SMC. By moving the switching to the higher derivatives of the control, chattering is significantly attenuated. Furthermore, as the order of the sliding mode
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
400
increases, precision increases with respect to the measurement time step whereas the classical (first order) sliding mode precision is proportional to the measurement step. This paper proposes sliding mode launch vehicle flight controls using classical and higher-order multiple and single loop designs. A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is shown. The intensive simulations demonstrate efficacy of the proposed control higher-order SMC algorithms for launch vehicle attitude control. 2. Launch vehicle attitude control mathematical model and problem formulation The equations of motion for a rigid body in the body frame with respect to the NorthEast-Down (NED) frame needed to completely control the attitude of a launch vehicle describe the vehicle kinematics and dynamics. The kinematics equations in quaternions are 1 q_ 0 ¼ qxT 2 1 q_ ¼ Kq x, 2 where q ¼ fq0 ,q1 ,q2 ,q3 gT vector 2 q0 q3 6 q q Kq ¼ 4 3 0 q2 q1
ð1Þ is a quaternion vector, q ¼ fq1 ,q2 ,q3 gT is a truncated quaternion q2
3
q1 7 5, q0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21 þ q22 þ q23 rbo1,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21 þ q22 þ q23 rbo1,
qffiffiffiffiffiffiffiffiffiffiffi q0 Z 1b2 ,
2 3 p 6 7 x¼4q5 r
ð2Þ
with p,q,r are roll, pitch and yaw rates, respectively. The dynamics equation is 1 x_ ¼ J1 0 XJ0 x þ J0 T þ FðÞ
where
2
0 6 r X¼4 q
3 2 Jxx q 6 J p 7 5, J ¼ 4 xy Jxz p 0 r 0
ð3Þ 3
Jxy
Jxz
Jyy Jyz
Jyz 7 5 is a matrix of inertia, and FðÞ 2 R3 is Jzz
a norm-bounded perturbation vector. The control problem for a launch vehicle is to determine the control torque command vector T in Eq. (3) such that the commanded orientation profiles qc are robustly asymptotically followed lim :qc ðtÞqðtÞ: ¼ 0
t- 1
ð4Þ
in the presence of the bounded disturbance torque, launch vehicle inertia variations, and aerodynamic surface and engine failures that are contained in FðÞ 2 R3 . Note that it is
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
401
sufficient to track the reduced quaternion q. It is assumed that the measured quaternion and body rate vectors are well estimated. 3. Background of higher order sliding mode control Sliding mode control emerged as the frontrunner of a particular type of control system called a Variable Structure Control System (VSCS). Work on variable structure control systems was pioneered in Russia in the early 1960s, and it did not appear outside Russia until a book by Itkis [18] and a survey paper by Utkin [19] were published in English. Traditional sliding mode control (SMC) can handle relative degree ‘‘one’’ of input–output dynamics with respect to a special output, names a sliding variable. The higher-order sliding mode (HOSM) approach [12–14,20,21,23] considers dynamics system x_ ¼ aðt,xÞ þ bðt,xÞu s ¼ sðt,xÞ
ð5Þ
n
where x 2 R , u 2 R, and a,b are smooth vector fields, and s 2 R is a sufficiently smooth function (named sliding variable). The relative degree r of the system is assumed to be constant and known, and the zero dynamics are assumed to be stable. The task is to fulfill the constraint sðt,xÞ ¼ 0 in finite time and to keep it exactly by some feedback in the presence of the bounded disturbance aðt,xÞ. Since the relative degree of system (5) is equal to r, the input–output dynamics can be presented as sðrÞ ¼ hðt,xÞ þ gðt,xÞu
ð6Þ
It is assumed that: (A1) the terms hðt,xÞ,gðt,xÞ are unknown but bounded 9hðt,xÞ9rC40,
0oKm rgðt,xÞrKM
ð7Þ
(A2) system (5) is of minimum-phase.
It is worth noting that if sðt,xÞ ¼ 0 is achieved in finite time than, due to Eqs. (5) and (6), then s ¼ s_ ¼ s€ ¼ . . . ¼ sðr1Þ ¼ 0
ð8Þ
is achieved in finite time as well. Definition [12]. It is said that system (6) is in the r-th order sliding mode if Eq. (8) is fulfilled. HOSM control works by forcing the discontinuous control to act on the higher-order time derivatives of the sliding variable, instead of working on the first time derivative as happens in classical sliding mode control. Furthermore, as the order increases, precision of the sliding variable stabilization increases proportional to Dtr [21], where r is relative degree, or the order of the first discontinuous total time derivative of the sliding variable. Several HOSM controllers that drive system (6) to the r-th order sliding mode are proposed. They include the twisting and the super-twisting second order sliding mode
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
402
(2-SMC) control [20,21], the nested HOSM control [12], and the quasi-continuous HOSM control [14]. The quasi-continuous HOSM algorithm [14] is of particular importance, since it provides for finite-time stable r-th sliding motion on the manifold sðt,xÞ ¼ 0 by means of continuous control everywhere except this manifold. The quasi-continuous HOSM controls are constructed as follows: Let bi be positive for all i ¼ 1,r1 and Ni,r positive definite. Then, in accordance with [14] _ s,. € . .,sðr1Þ Þ u ¼ u0 Cr1,r ðs, s,
ð9Þ
is r-sliding homogeneous and provides finite time stability where ðriÞ=ðriþ1Þ j0,r ¼ s, N0,r ¼ 9s9, C0,r ¼ j0,r =N0,r ¼ signs, ji,r ¼ sðiÞ þ bi Ni1,r Ci1,r , Ni,r ¼ ðriÞ=ðriþ1Þ 9sðiÞ 9þ bi Ni1,r , Ci,r ¼ ji,r =Ni,r , and u0 are large enough. The control (9) is a continuous function of time everywhere except the r-sliding set (8). Examples of 1st, 2nd and 3rd quasi-continuous HOSM control laws are as follows [14]: r¼1 :
u ¼ u0 signðsÞ
r¼2 :
u ¼ u0
r¼3 :
u ¼ u0
s_ þ 9s9
ð10Þ 1=2
signðsÞ
s_ þ 9s9
ð11Þ
1=2
_ þ 9s9 s€ þ 2ð9s9
2=3 1=2
€ þ 2ð9s9 _ þ 9s9 9s9
Þ
ðs_ þ 9s9
2=3 1=2
Þ
2=3
signðsÞÞ
2=3
9s_ þ 9s9
ð12Þ
signðsÞ9
_ s,. € . .,sðr1Þ are required, Since the real-time exact calculation or measurements of s, s, the quasi-continuous control can be used with the exact higher-order sliding mode differentiators [12,15]. 8 z_ 0 ¼ v0 , > > > > ðr1Þ=r > > v0 ¼ lr L1=r 9z0 sðtÞ9 signðz0 sðtÞÞ þ z1 > > > > > _ ¼ v , z > 1 1 > > > < v1 ¼ lr1 L1=ðr1Þ 9z1 v0 9ðr2Þ=ðr1Þ signðz1 v0 Þ þ z2 ð13Þ > ^ > > > > > z_ r2 ¼ vr2 , > > > > ð1=2Þ > > signðzr2 vr3 Þ þ zr v_ r2 ¼ l2 L1=2 9zr2 vr3 9 > > > : z_ ¼ l Lsignðz v Þ r1
1
r1
r2
_ where z0 -sðtÞ, z1 -sðtÞ,. . .,zr1 -sðr1Þ ðtÞ in finite time. The effect of measurement noise on the accuracy of the HOSM differentiator (13) is estimated [12]. The parameters of the differentiator (13) are to be chosen in accordance with the condition 9sðrÞ 9rL, and L is to satisfy L4C þ u0 KM [12–14]. The sequence of the gains l1 ,l2 ,. . .,lr1 can be chosen in advance [12]. Therefore, in the case when C,Km ,KM are known, only the parameter u0 is needed to be tuned. Usually both L and u0 can be determined via computer simulation.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
403
4. Classical and higher order sliding mode launch vehicle attitude control designs Both multiple loop and single loop higher order sliding mode launch vehicle attitude controllers of varying orders are shown using the quasi-continuous higher order algorithms shown in Eqs. (9)–(13). For the multiple-loop, the approach taken is that of a two-loop structure based on the theory of singular perturbation. We assume that the launch vehicle generates body rate commands slower than the torque commands are produced, which are then allocated and actuated by the aero-surfaces and/or engines of the launch vehicle. Upon this assumption, we subsequently divide the systems into two subsystems, one corresponding to slower dynamics and other corresponding to faster dynamics, and then design controllers for each one of them separately. Thus, the multipleloop sliding mode control design is composed of an outer loop, which generates body rate commands, and the inner loop which produces the torque commands. The single loop design takes two different approaches. The first is accomplished by generating the body rate that are calculating directly from the given in current time quaternion commands. The other approach is that of a higher-order quasi-continuous control using control T as the design variable. With the single loop control, time-scaling [11] no longer needs to be preserved. 4.1. Multiple loop sliding mode attitude control driven by sliding mode disturbance observer (SMC-SMDO) The smooth multiple-loop classical SMC-SMDO sliding mode controller [11] is designed in three steps: Step 1: The outer (guidance) loop the SMC-like smooth controller generates body rate commands (virtual control) _ xc ¼ 2K1 q ½qc þ K1 eq þ K2 s0 ) xc ¼ 2K-1 ½q_ c þ ðK1 þ K2 Þeq þ K2 K1 $c , q
where
Z
ð14Þ
t
eq dt,
s0 ¼ eq þK1
$ _ c ¼ eq
s0 2 R3 ,
eq ¼ qc q
ð15Þ
0
that provide the following outer loop (guidance) compensated dynamics (given the xc profile is tracked perfectly in the inner loop) e€ q þðK1 þ K2 Þ_e q þðK2 K1 Þeq ¼ 0,
eq ð0Þ ¼ eq0 ,
e_ q ð0Þ ¼ e_ q0
ð16Þ
Remark 1. The sliding variable s0 in (15) is driven to zero asymptotically by the SMC-like smooth virtual control xc in (14). The quaternion tracking error eq dynamics obey the second order linear homogeneous differential Eq. (16). It means that the output tracking error eq tends to zero as time increases upon a corresponding selection of the diagonal matrices K1 and K2 . Step 2: The inner (faster) loop smooth sliding mode controller driven by a sliding mode disturbance observer (SMC-SMDO) generates control torque commands T ¼ T1 þ T2
ð17Þ
404
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
where T1 compensates for the known RLV dynamics and provides for the desired sliding variable, so , stabilization dynamics as well as the desired overall body rate tracking error dynamics. This is _ c þJ0 K3 ex þ OJ0 x þ J0 K4 sx ) T1 ¼ J0 x _ c þOJ0 xc þ ðJ0 K3 OJ0 þ J0 K4 Þex þ J0 K4 K3 ge , T1 ¼ J0 x
g_ e ¼ ex
ð18Þ
and T2 is generated using a corresponding SMDO, which design will be presented in Step 3, to compensate for the perturbation term FðÞ in Eq. (3) ^ T2 ¼ J0 FðÞ ð19Þ The sliding variable, so , is designed Z t so ¼ eo þK3 eo dt, so 2 R3 , eo ¼ oc o
ð20Þ
0
and is driven to zero asymptotically via the SMC-SMDO (17)–(19). The following body rate compensated dynamics (given exact estimation of the perturbation term in (3) by ^ ¼ FðÞ and the T profile is allocated perfectly into commands to actuator SMDO, i.e. FðÞ deflections) is provided in the inner loop: e€ x þðK3 þK4 Þ_e x þK4 K3 ex ¼ 0,
ex ð0Þ ¼ ex0 ,
e_ x ð0Þ ¼ e_ x0
ð21Þ
Step 3: Sliding mode disturbance observer (SMDO) is designed for estimating the body rate dynamics in Eq. (3) that is rewritten in a form: x_ ¼ Uð Þþu
ð22Þ
-1 where Uð Þ ¼ J1 0 ðXJ0 xþFðÞÞ and u ¼ J0 T. Eq. (22) can be rewritten in a scalar format
o_ i ¼ FðÞi þ ui ,
8i ¼ 1,2,3
Auxiliary sliding variables, s~ i , are introduced ( s~ i ¼ oi zi z_ i ¼ ui vi
ð23Þ
ð24Þ
and are driven to zero in finite time by means of the classical SMC injection term vi ¼ ri signð~s i Þ,
ri 49Fi ðÞ9
ð25Þ
Filtering the injection terms (25) in the sliding modes s~ i ¼ 0 by a low pass filter (LPF) of a desired relative degree we obtain estimates of FðÞi ^ i ¼ LPF fvi g FðÞ
ð26Þ
Finally, the perturbation term FðÞ is reconstructed: ^ ¼ J0 FðÞ ^ þ OJ0 o FðÞ
ð27Þ
Remark 2. The sliding variable so in (20) is driven to zero asymptotically by the smooth SMC-SMDO control torque in Eqs. (18), (19), and (27). The body rate tracking error dynamics ex obey the second order linear homogeneous differential equation (21). It means that the output tracking error ex tends to become zero as time increases upon a corresponding selection of the diagonal matrices K3 and K4 . Writing inner and outer loop
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
405
tracking error equations in a damping ratio/natural frequency format 2 € _ xþ2nx n xþx nx ¼ 0
ð28Þ
it is easy to calculate the elements of the diagonal matrices K1, K2 , K3 and K4 to provide given damping factors (usually all are equal to 1.1 or so) and natural frequencies that provide for a sufficient time-scale separation between the control loops to Eqs. (16) and (21) for inner and outer loop compensated tracking dynamics. 4.2. Multiple loop Higher order sliding mode attitude control The outer loop HOSM controller generates body rate commands. The launch vehicle motion that is considered in the outer loop is described in quaternions in Eq. (1). The launch vehicle motion that is considered in the inner loop is described by Eq. (2). The control torque T is considered as a control that will be allocated into aerodynamic surface deflection commands, rocket engine chamber pressure commands, and reaction control system commands. Since the control action xc that is fed to an inner loop may be required to be smooth for actuator input, a third order outer loop is developed with a second order inner loop. The double loop continuous HOSM controller is designed in two steps. Step 1: For the outer loop, we let r0 ¼ qc q 2R3 , and we wish to drive r0 -0 in finite time. In order to achieve a sufficient smoothness of the body rate command xc ðtÞ, the outer € loop quasi-continuous HOSM control law is designed in terms of v2 ¼ x. 1 Denoting x ¼ 2Kq x, the sliding variable dynamics are derived 8 _ > < r_ 0 ¼ qc þx ð3Þ _ ¼ v1 x ð29Þ -rð3Þ 0 ¼ qc þ v 2 > : v_ 1 ¼ v2 Relative degree of the outer loop sliding variable r0 -0 is equal to 3. Therefore, the outer loop 3rd order quasi-continuous HOSM control law is designed xc ¼ 2K1 q x € ¼ v2 x v_ 2j ¼ a2j
s€ oj þ 2ð9s_ oj 9 þ 9soj 9
2=3 1=2
Þ
ðs_ oj þ 9soj 9
2=3 1=2
9s€ oj 9 þ 2ð9s_ oj 9 þ 9soj 9
Þ
2=3
9s_ oj þ 9soj 9
signðsoj ÞÞ
2=3
,
j ¼ 1,2,3
ð30Þ
signðsoj Þ9
By making a2j 40, j ¼ 1,2,3 large enough, the 3rd order HOSM control law control law (30) will drive r0 , r_ 0 , r€ 0 -0 in finite time. Step 2 For the inner loop: we let r1 ¼ xc x, and we wish to drive r1 -0 in finite time. In order to achieve sufficient smoothness of the torque command TðtÞ, the inner loop _ Denoting T ¼ quasi-continuous SOSM control law is designed in terms of v3 ¼ T. 1 1 J0 XJ0 x þ J0 T, the sliding variable r1 dynamics are derived ( _ c Fð ÞT r_ 1 ¼ x _ Þv3 € c Fð ð31Þ -r€ 1 ¼ x T_ ¼ v3
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
406
and the inner loop quasi-continuous 2-SMC control law is designed as T ¼ J0 TXJ0 x T_ ¼ v 3
v3j ¼ a3j
s_ 1j þ 9s1j 9
1=2
signðs1j Þ
9s_ 1j 9 þ 9s1j 9
1=2
j ¼ 1,2,3,
,
ð32Þ
By making a3j 40, j ¼ 1,2,3 large enough, the SOSM control law (27) will drive r1 , r_ 1 -0 in finite time. In order to implement the HOSM double loop controller (30) and (32) the sliding variable derivatives r_ 0 , r€ 0 , r_ 1 can be obtained via HOSM exact sliding mode differentiator (13). Remark 3. The designed double-loop HOSM controller is continuous and drives the quaternion tracking error to zero in finite time with its derivative. Therefore, full dynamical collapse is achieved via smooth control in systems with the bounded disturbance. In theory, we can continue creating arbitrary order sliding mode controls using this technique, which can achieve higher orders of accuracy. However, depending upon the application, higher orders of sliding mode control design may be difficult to implement. 4.3. Single loop Higher order sliding mode attitude control The same finite-convergence time quaternion tracking can be achieved by designing a single loop HOSM controller using the quasi-continuous higher order sliding mode control technique. The proposed single loop control structure is simpler than the double-loop one presented in Section 4.1. We let rs ¼ qc q, and we wish to drive rs -0 in finite time. In order to achieve the continuity of the torque command Tc ðtÞ, the single-loop quasi-continuous SOSM control _ Denoting T ¼ 1K ðJ1 XJ x þ J1 TÞ, the sliding law is designed in terms of v4 ¼ T. 0 0 2 q 0 variable rs dynamics are derived as 8 > < r€ s ¼ q€ þ 1 K_ q xþ 1 Kq Fð ÞþT 1_ 1_ 1 _ c ð3Þ 1 € 2 2 _ Þþv4 -rð3Þ K q Fð Þ þ Kq Fð s ¼ qc þ K q xþ K q xþ > 2 2 2 2 : T_ ¼ v 4
ð33Þ and the single-loop quasi-continuous SOSM control law is designed. This is T ¼ 2J0 K1 q TXJ0 x T_ ¼ v4 v_ 4j ¼ a4j
s€ sj þ 2ð9s_ sj 9 þ 9ssj 9
2=3 1=2
9s€ oj 9 þ 2ð9s_ sj 9 þ 9ssj 9
Þ
ðs_ sj þ 9ssj 9
2=3 1=2
Þ
2=3
9s_ sj þ 9ssj 9
signðssj ÞÞ
2=3
,
j ¼ 1,2,3
ð34Þ
signðssj Þ9
By making a4j 40, j ¼ 1,2,3 large enough, the control law (34) will drive rs , r_ s , r€ s -0 in finite time. That means achieving the exact tracking of the commanded quaternion profiles, or full dynamical collapse.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
407
Remark 4. It is worth noting that relative degree of system can be increased should we take into account the dynamics of the actuator. If the dynamics of the actuator are considered as parasitic ones, a limit cycle could occur and system’s stability must be re-evaluated [23].
5. Simulations A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is presented. The flight control system of RSLV SLV-X based on single-loop and double-loop HOSM controller is simulated using Matlab/Simulink Version 6.5 with Aerospace Blockset. The launch vehicle model used is a high fidelity 6DOF model with table look up aerodynamic and torque coefficients and a realistic wind model. The model also simulates natural environments. The atmospheric model uses the Committee on Extension to the Standard Atmosphere (COESA) United States standard atmospheric values. The Gravity Model implements the geocentric equipotential ellipsoid of the World Geodetic System (WGS84). The Kennedy Space Center Wind Model and Dryden Wind Turbulence Model blocks implement the mathematical representation of wind disturbance. The simulation trajectories of the RSLV SLV-X are available up to 400 s. For testing of the controllers, 126 s of flight time was used, which corresponds to a staging event that occurs, and allows time to show recovery after that staging event. Winds were also used for selective plots. The wind disturbances are highest at the very beginning, and then tend to settle to zero at about 70 s into flight. Fig. 2 shows the wind angular rate disturbances (in degrees/s) acting on the vehicle. We see that peak winds reach about 10 1/s just after liftoff, and settle down at about 63 s. After about 105 s, which corresponds to about 120,000 ft altitude, wind disturbances are essentially zero. The results of the simulations using single and double-loop HOSM 12
Angular rate - q (degrees)
10 8 6 4 2 0 -2 -4 -6 -8 0
20
40
60
80
Time (sec) Fig. 2. Winds profile.
100
120
140
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
408
100 Pitch Cmd Pitch Sensed
90
Pitch (degrees)
80 70 60 50 40 30 20 0
20
40
60
80 Time (sec)
100
120
140
120
140
Fig. 3. Pitch angle tracking (HOSM single loop).
0.015
Pitch Error (degrees)
0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 0
20
40
60 80 Time (sec)
100
Fig. 4. Pitch tracking error (HOSM single loop).
controllers appeared to be very similar, and the presented plots are for the single loop controller only. Fig. 3 shows the pitch tracking (in degrees) via single-loop HOSM controller. One can see that the HOSM single loop tracks so accurate in the presence of the disturbance (Fig. 2) that it is hard to see a difference in the command vs. the sensed. The pitch tracking error is shown in Fig. 4. It is clear that on average the error, and also the corresponding sliding variable, is kept at zero. There are time instants that controller moves off the surface, but it recovers almost instantly. We see that the highest error magnitude is at 0.021.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
409
Pitch Error Derivative (degrees/sec)
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0
20
40
60 80 Time (sec)
100
120
140
120
140
Pitch Error 2nd Derivative (degrees/sec2)
Fig. 5. Pitch error derivative (HOSM single loop). 400 200 0 -200 -400 -600 -800 0
20
40
60 80 Time (sec)
100
Fig. 6. Pitch error 2nd derivative (HOSM single loop).
Fig. 5 shows the first time derivative of the pitch angle tracking error (that is proportional to the derivative of the corresponding sliding variable). We see that the single loop HOSM controller drives the first derivative of our variable to zero in finite time, and does very nicely in maintaining the derivative at zero in spite of the disturbances. The HOSM single loop controller also drives the 2nd time derivative of our sliding variable to zero in finite time as shown in Fig. 6. Figs. 7 and 8 show the effect of wind disturbances on pitch angular rate as well as the control torque. Fig. 9 shows a comparison of the pitch rate profiles between the traditional double loop SMC-SMDO [10,11] and HOSM single loop controller in the pitch channel. Wind disturbances were removed to more accurately show the differences in the angular rates.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
410
12 10
Pitch Rate (degrees/sec)
8 6 4 2 0 -2 -4 -6 -8 0
20
40
60 80 Time (sec)
100
120
140
120
140
Fig. 7. Pitch rate (HOSM single loop).
6
x 105
Pitch Torque (lbs ft)
4 2 0 -2 -4 -6 -8 0
20
40
60 80 Time (sec)
100
Fig. 8. Pitch control torque (HOSM single loop).
As we can see, even without the disturbances, the HOSM enforces pitch rate is much more aggressive. This is due to the fact that the use of the HOSM single loop algorithm foregoes controlling of the angular rates in order to provide more robustness. This is desirable only in situations where the launch vehicle is not carrying a human payload. There are angular rate restrictions on humans in space flight that would require angular rate control available in the double loop but not in a single loop. This would also be applicable to reusable launch vehicles in certain configurations. The single loop shown here is desirable only for expendable uncrewed launch vehicles such as the FALCON RSLV SLV-X.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
411
Fig. 9. Comparison of body pitch rates: SMC-SMDO and HOSM Single Loop.
6. Conclusion The Higher Order Sliding Mode Attitude controls for launch vehicles for the multiple and single loop architectures are shown and tested on a simulation platform based on SLV-X Launch Vehicle. These controllers allow for customization of launch vehicle attitude controls depending on the type of vehicle used. Unmanned ELVs do not return to Earth and normally employ a high control authority. Furthermore, control authority can be fully utilized when the ELV is not human rated since the limits on angular rates for an unmanned flight are higher. Therefore the control approach for unmanned ELVs is to provide a simple, but robust, control algorithm like the single order HOSM that will utilize the full control authority of the launch vehicle. Since human rated launch vehicles require less aggressive angular rate profiles due to crew being onboard, a control approach that will minimize the angular rates but still provide adequate robustness is needed, in particular the SMC-SMDO double loop flight controller. References [1] John Hanson, Integration and Testing of Advanced Guidance and Control Technologies, In: Proceedings of the Advanced Guidance and Control Workshop, NASA/MSFC, July 2003. [2] C.E. Hall, A.S. Hodel, J.Y. Hung, Variable structure PID control to prevent integral windup, in: Proceedings of the IEEE 31st Southeastern Symposium on System Theory, 1999, pp. 169–173. [3] C.E. Hall, M.W. Gallaher, N.D. Hendrix, X-33 attitude control system design for ascent, transition, and entry flight regimes, in: Proceedings of AIAA Guidance, Navigation and Control Conference, AIAA-1998-4411, 1998. [4] J. Zhu, A. Scott Hodel, K. Funston, C.E. Hall, X-33 entry flight controller design by trajectory linearization—a singular perturbation approach, in: Proceedings of American Astronautical Society Guidance and Control Conference, Breckenridge, CO, January 2001, pp. 151–170.
412
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
[5] A. Huizenga, T. Bevacqua, J. Fisher, D. Cooper, J. Zhu, Improved trajectory linearization flight controller for reusable launch vehicles, in: Proceedings of 42nd AIAA Aerospace Science Meeting, Reno, NV, AIAA 2004-0875, January 2004. [6] M. Johnson, T. Calise, E. Johnson, Evaluation of an adaptive method for launch vehicle flight control, in: Proceedings of AIAA Guidance, Navigation and Control Conference, Austin, TX, August 2003. [7] E. Johnson, T. Calise, H. El-Shirbiny, Feedback linearization with neural network augmentation applied to X-33 attitude control, in: Proceedings of AIAA Guidance, Navigation and Control Conference, 2000. [8] V. Utkin, J. Guldner, J. Shi, Sliding Modes in Electromechanical Systems, Taylor and Francis, London, 1999. [9] C. Edwards, S. Spurgeon, Sliding Mode Control, Taylor & Francis, Bristol, PA, 1998. [10] Y. Shtessel, J. Stott, J. Zhu, Time-varying sliding mode controller with sliding mode observer for reusable launch vehicles, in: Proceedings of AIAA Guidance, Navigation and Control Conference, Austin, Texas, August 2003. [11] Y. Shtessel, J. Stott, J. Zhu, Reusable launch vehicle control via time-varying multiple time scale sliding modes, in: Proceedings of the 17th IMACS World Congress Scientific Computation, Applied Mathematics and Simulation, Paris, France, July 2005. [12] A. Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control 76 (9/10) (2003) 924–941. [13] A. Levant, Construction principles of 2-sliding mode design, Automatica 43 (4) (2007) 576–586. [14] A. Levant, Quasi-continuous high-order sliding-mode controllers, IEEE Transactions on Automatic Control 50 (11) (2005) 1812–1816. [15] L. Fridman, Y. Shtessel, C. Edwards, X. Yan, Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems, International Journal of Robust and Nonlinear Control 18 (2008) 399–412. [16] C. Hall, Y. Shtessel, Sliding mode disturbance observers-based control for a reusable launch vehicle, AIAA Journal on Guidance, Control, and Dynamics 29 (6) (2006) 1315–1329. [17] Y. Shtessel, C. Hall, M. Jackson, Reusable launch vehicle control in multiple time scale sliding modes, AIAA Journal on Guidance Control, and Dynamics 23 (6) (2000) 1013–1020. [18] U. Itkis, Control Systems of Variable Structure, Keter Publishing house, Jerusalem, 1976. [19] V.I. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control 22 (1977) 212–222. [20] S.V. Emelyanov, S.K. Korovin, L.V. Levantovsky, Higher order sliding regimes in the binary control systems, Soviet Physics, Doklady 31 (1986) 291–293. [21] A. Levant (L.V. Levantovsky), Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (1993) 1247–1263. [22] D. Drake, M. Xin, S.N. Balakrishnan, A new nonlinear control technique for ascent phase of reusable launch vehicles, AIAA Journal of Guidance, Control and Dynamics 27 (6) (2004) 938–948. [23] I. Boiko, L. Fridman, A. Pisano, E. Usai, Analysis of chattering in systems with second-order sliding modes, IEEE Transactions on Automatic Control 52 (11) (2007) 2085–2101. [24] M.C. Pai, Design of adaptive sliding mode controller for robust tracking and model following, Journal of the Franklin Institute 347 (10) (2010) 1837–1849. [25] H. Yang, Y. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via delta operator approach, Journal of the Franklin Institute 347 (7) (2010) 1199–1213. [26] I. Boiko, Frequency domain precision analysis and design of sliding mode observers, Journal of the Franklin Institute 347 (6) (2010) 899–909. [27] M. Basin, D. Calderon-Alvarez, Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems, Journal of the Franklin Institute 347 (6) (2010) 910–922. [28] Y. Xia, Z. Zhu, C. Li, H. Yang, Q. Zhu, Robust adaptive sliding mode control for uncertain discrete-time systems with time delay, Journal of the Franklin Institute 347 (1) (2010) 339–357.
Journal of the Franklin Institute 349 (2012) 397–412 www.elsevier.com/locate/jfranklin
Launch vehicle attitude control using sliding mode control and observation techniques James E. Stotta, Yuri B. Shtesselb,n a
Safety and Mission Assurance Directorate, QD33, NASA, Marshall Space Flight Center, AL 35812, USA b Department of Electrical and Computer Engineering, The University of Alabama in Huntsville, Huntsville, AL 35899, USA Received 2 March 2011; received in revised form 6 June 2011; accepted 28 July 2011 Available online 16 August 2011
Abstract In determining flight controls for launch vehicle systems, several uncertain factors must be taken into account, including a variety of payloads, a wide range of flight conditions and different mission profiles, wind disturbances and plant uncertainties. Crewed vehicles must adhere to human rating requirements, which limit the angular rates. Sliding mode control algorithms that are inherently robust to external disturbances and plant uncertainties are very good candidates for improving the robustness and accuracy of the flight control systems. Recently emerging Higher Order Sliding Mode (HOSM) control is even more powerful than the classical Sliding Mode Controls (SMC), including the capability to handle systems with arbitrary relative degree. This paper proposes sliding mode launch vehicle flight controls using classical SMC driven by the sliding mode disturbance observer (SMDO) and higherorder multiple and single loop designs. A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is shown. The intensive simulations demonstrate efficacy of the proposed HOSM and SMC-SMDO control algorithms for launch vehicle attitude control. & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
1. Introduction Launch Vehicles, in general terms, are vehicles with the primary purpose of carrying a payload from the surface of the Earth into outer space. Outer space, in contrast to air space, is most commonly defined as everything beyond the Karman line 100 km n
Corresponding author. Tel.:þ12568246164; fax:þ12568246803. E-mail addresses: [email protected] (J.E. Stott), [email protected] (Y.B. Shtessel).
0016-0032/$32.00 & 2011 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2011.07.020
398
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
Nomenclature q ¼ fq0 ,q1 ,q2 ,q3 gT quaternion vector x ¼ fp,q,rgT body rate vector (rad/s) J matrix of inertia sðÞ ,sðÞ sliding variables T control torque (moment) (lbs-ft) Fð Þ the bounded disturbance that contains the disturbance torque, launch vehicle inertia variations, and aerodynamic surface and engine failures
(62 miles) above the Earth’s surface. This line roughly marks the altitude at which the density of the atmosphere of the Earth is so low that a vehicle would have to fly faster than orbital speed, i.e., the speed tangential to the Earth’s surface, which is required to maintain an orbit of the Earth at that altitude, in order to support flight using aerodynamic lift. Launch Vehicles can support flight at altitudes above the Karman line since they use thrust instead of aerodynamic lift to overcome the force of gravity. Launch Vehicles are generally classified as either Expendable or Reusable (or partially reusable), orbital or sub-orbital, and manned or unmanned. Expendable Launch Vehicles (ELVs) are designed to be for one-time use and normally fall back to Earth once the payload is deployed breaking apart in the Earth’s atmosphere. Reusable Launch Vehicles (RLVs), or reusable components of Partially Reusable Launch Vehicles (P-RLVs), are designed to perform many missions within their life cycle. Orbital Launch Vehicles can carry a payload into a stable orbit about the Earth, whereas a suborbital Launch Vehicle cannot. Other characterizations of Launch Vehicles are mass to orbit ratios, number of stages, and type of propulsion (e.g., solid or liquid). Examples of several types of launch vehicles are shown in Fig. 1. As Launch Vehicles continue to evolve along with the development of new technologies, the main objectives of new designs are to increase performance, reliability, safety, and availability, while decreasing cost. Technologies such as structures, avionics, health management systems, and ground operations are studied to determine optimal ways of reducing cost and risks associated with Launch Vehicle missions. As computational capability increases, control algorithms play an ever more important role as higher levels of performance, reliability, safety, and availability are achieved. Currently, the most ubiquitous control algorithm used is the Proportional Integral Derivative (PID) control with gain scheduling. PID control was used on the Saturn V and the Space Shuttle, and is being developed in the Ares rockets. Although many updates have taken place over the last 30 years, PID with gain scheduling control, remains dominant due to its proven heritage. In the late 1990s, NASA’s Marshall Space Flight Center (MSFC) led an Advanced Guidance and Control (AG&C) project funded by the X-33 Program Office to study ascent and entry controllers. This work continued under the Integration and Testing of Advanced Guidance and Control Technologies (ITAGCT) for the Space Launch Initiative (SLI) Program. The goal of the flight control system concepts that were developed under NASA’s AG&C project was to seek new launch vehicle control technologies to successfully fly in an expanded range of flight conditions, vehicle parameter variations, mission profiles, and unplanned failures requiring in-flight adaptation, as well
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
399
Fig. 1. Launch vehicles.
as to significantly reduce the number of Guidance, Navigation, and Control (GN&C) tuning parameters [1]. Several control techniques, including Proportional-Integral-Derivative (PID) control law with gain scheduling [2,3], Trajectory Linearization Control (TLC) [4,5], adaptive Neural Network (NN) [6,7], and Theta-D [22] launch vehicle control algorithms have been developed under AG&C and used for launch vehicle flight control. However, the robustness issues are still not fully addressed by means of these control techniques. The Sliding Mode Control (SMC) algorithm [8,9,24–28], is a Lyapunov function based technique, providing global stability thus avoiding gain scheduling. It is an attractive design because of its inherent insensitivity and robustness to unknown plant uncertainties and external disturbances. Furthermore, its robustness accommodates different trajectories and aerodynamic surface and engine failures without gain scheduling. Due to these advantages over the PID and other controllers, the SMC is believed to reduce risk and dramatically decrease the amount of time spent in pre-flight analysis, thus reducing cost over current values [10,11,16,17]. Recently emerging Higher Order Sliding Mode (HOSM) control [12–15,23] is even more powerful than the classical SMC, including the capability to handle systems with arbitrary relative degree that allows avoiding multiple loop control system design with partial or full dynamical collapse (i.e. reduction of the order of the compensated dynamics). By increasing the system’s relative degree artificially, sliding control of arbitrarily smooth order can be achieved, which completely removes the chattering effect. This HOSM approach allows achieving control smoothness without approximation of sign function by a saturation one (as it often done in classical SMC), while retaining the main properties of the classical SMC and improving its accuracy. HOSM works with the discontinuous control acting on the higher-order time derivatives of the sliding variable, instead of working on the first time derivative as happens in classical SMC. By moving the switching to the higher derivatives of the control, chattering is significantly attenuated. Furthermore, as the order of the sliding mode
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
400
increases, precision increases with respect to the measurement time step whereas the classical (first order) sliding mode precision is proportional to the measurement step. This paper proposes sliding mode launch vehicle flight controls using classical and higher-order multiple and single loop designs. A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is shown. The intensive simulations demonstrate efficacy of the proposed control higher-order SMC algorithms for launch vehicle attitude control. 2. Launch vehicle attitude control mathematical model and problem formulation The equations of motion for a rigid body in the body frame with respect to the NorthEast-Down (NED) frame needed to completely control the attitude of a launch vehicle describe the vehicle kinematics and dynamics. The kinematics equations in quaternions are 1 q_ 0 ¼ qxT 2 1 q_ ¼ Kq x, 2 where q ¼ fq0 ,q1 ,q2 ,q3 gT vector 2 q0 q3 6 q q Kq ¼ 4 3 0 q2 q1
ð1Þ is a quaternion vector, q ¼ fq1 ,q2 ,q3 gT is a truncated quaternion q2
3
q1 7 5, q0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21 þ q22 þ q23 rbo1,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q21 þ q22 þ q23 rbo1,
qffiffiffiffiffiffiffiffiffiffiffi q0 Z 1b2 ,
2 3 p 6 7 x¼4q5 r
ð2Þ
with p,q,r are roll, pitch and yaw rates, respectively. The dynamics equation is 1 x_ ¼ J1 0 XJ0 x þ J0 T þ FðÞ
where
2
0 6 r X¼4 q
3 2 Jxx q 6 J p 7 5, J ¼ 4 xy Jxz p 0 r 0
ð3Þ 3
Jxy
Jxz
Jyy Jyz
Jyz 7 5 is a matrix of inertia, and FðÞ 2 R3 is Jzz
a norm-bounded perturbation vector. The control problem for a launch vehicle is to determine the control torque command vector T in Eq. (3) such that the commanded orientation profiles qc are robustly asymptotically followed lim :qc ðtÞqðtÞ: ¼ 0
t- 1
ð4Þ
in the presence of the bounded disturbance torque, launch vehicle inertia variations, and aerodynamic surface and engine failures that are contained in FðÞ 2 R3 . Note that it is
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
401
sufficient to track the reduced quaternion q. It is assumed that the measured quaternion and body rate vectors are well estimated. 3. Background of higher order sliding mode control Sliding mode control emerged as the frontrunner of a particular type of control system called a Variable Structure Control System (VSCS). Work on variable structure control systems was pioneered in Russia in the early 1960s, and it did not appear outside Russia until a book by Itkis [18] and a survey paper by Utkin [19] were published in English. Traditional sliding mode control (SMC) can handle relative degree ‘‘one’’ of input–output dynamics with respect to a special output, names a sliding variable. The higher-order sliding mode (HOSM) approach [12–14,20,21,23] considers dynamics system x_ ¼ aðt,xÞ þ bðt,xÞu s ¼ sðt,xÞ
ð5Þ
n
where x 2 R , u 2 R, and a,b are smooth vector fields, and s 2 R is a sufficiently smooth function (named sliding variable). The relative degree r of the system is assumed to be constant and known, and the zero dynamics are assumed to be stable. The task is to fulfill the constraint sðt,xÞ ¼ 0 in finite time and to keep it exactly by some feedback in the presence of the bounded disturbance aðt,xÞ. Since the relative degree of system (5) is equal to r, the input–output dynamics can be presented as sðrÞ ¼ hðt,xÞ þ gðt,xÞu
ð6Þ
It is assumed that: (A1) the terms hðt,xÞ,gðt,xÞ are unknown but bounded 9hðt,xÞ9rC40,
0oKm rgðt,xÞrKM
ð7Þ
(A2) system (5) is of minimum-phase.
It is worth noting that if sðt,xÞ ¼ 0 is achieved in finite time than, due to Eqs. (5) and (6), then s ¼ s_ ¼ s€ ¼ . . . ¼ sðr1Þ ¼ 0
ð8Þ
is achieved in finite time as well. Definition [12]. It is said that system (6) is in the r-th order sliding mode if Eq. (8) is fulfilled. HOSM control works by forcing the discontinuous control to act on the higher-order time derivatives of the sliding variable, instead of working on the first time derivative as happens in classical sliding mode control. Furthermore, as the order increases, precision of the sliding variable stabilization increases proportional to Dtr [21], where r is relative degree, or the order of the first discontinuous total time derivative of the sliding variable. Several HOSM controllers that drive system (6) to the r-th order sliding mode are proposed. They include the twisting and the super-twisting second order sliding mode
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
402
(2-SMC) control [20,21], the nested HOSM control [12], and the quasi-continuous HOSM control [14]. The quasi-continuous HOSM algorithm [14] is of particular importance, since it provides for finite-time stable r-th sliding motion on the manifold sðt,xÞ ¼ 0 by means of continuous control everywhere except this manifold. The quasi-continuous HOSM controls are constructed as follows: Let bi be positive for all i ¼ 1,r1 and Ni,r positive definite. Then, in accordance with [14] _ s,. € . .,sðr1Þ Þ u ¼ u0 Cr1,r ðs, s,
ð9Þ
is r-sliding homogeneous and provides finite time stability where ðriÞ=ðriþ1Þ j0,r ¼ s, N0,r ¼ 9s9, C0,r ¼ j0,r =N0,r ¼ signs, ji,r ¼ sðiÞ þ bi Ni1,r Ci1,r , Ni,r ¼ ðriÞ=ðriþ1Þ 9sðiÞ 9þ bi Ni1,r , Ci,r ¼ ji,r =Ni,r , and u0 are large enough. The control (9) is a continuous function of time everywhere except the r-sliding set (8). Examples of 1st, 2nd and 3rd quasi-continuous HOSM control laws are as follows [14]: r¼1 :
u ¼ u0 signðsÞ
r¼2 :
u ¼ u0
r¼3 :
u ¼ u0
s_ þ 9s9
ð10Þ 1=2
signðsÞ
s_ þ 9s9
ð11Þ
1=2
_ þ 9s9 s€ þ 2ð9s9
2=3 1=2
€ þ 2ð9s9 _ þ 9s9 9s9
Þ
ðs_ þ 9s9
2=3 1=2
Þ
2=3
signðsÞÞ
2=3
9s_ þ 9s9
ð12Þ
signðsÞ9
_ s,. € . .,sðr1Þ are required, Since the real-time exact calculation or measurements of s, s, the quasi-continuous control can be used with the exact higher-order sliding mode differentiators [12,15]. 8 z_ 0 ¼ v0 , > > > > ðr1Þ=r > > v0 ¼ lr L1=r 9z0 sðtÞ9 signðz0 sðtÞÞ þ z1 > > > > > _ ¼ v , z > 1 1 > > > < v1 ¼ lr1 L1=ðr1Þ 9z1 v0 9ðr2Þ=ðr1Þ signðz1 v0 Þ þ z2 ð13Þ > ^ > > > > > z_ r2 ¼ vr2 , > > > > ð1=2Þ > > signðzr2 vr3 Þ þ zr v_ r2 ¼ l2 L1=2 9zr2 vr3 9 > > > : z_ ¼ l Lsignðz v Þ r1
1
r1
r2
_ where z0 -sðtÞ, z1 -sðtÞ,. . .,zr1 -sðr1Þ ðtÞ in finite time. The effect of measurement noise on the accuracy of the HOSM differentiator (13) is estimated [12]. The parameters of the differentiator (13) are to be chosen in accordance with the condition 9sðrÞ 9rL, and L is to satisfy L4C þ u0 KM [12–14]. The sequence of the gains l1 ,l2 ,. . .,lr1 can be chosen in advance [12]. Therefore, in the case when C,Km ,KM are known, only the parameter u0 is needed to be tuned. Usually both L and u0 can be determined via computer simulation.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
403
4. Classical and higher order sliding mode launch vehicle attitude control designs Both multiple loop and single loop higher order sliding mode launch vehicle attitude controllers of varying orders are shown using the quasi-continuous higher order algorithms shown in Eqs. (9)–(13). For the multiple-loop, the approach taken is that of a two-loop structure based on the theory of singular perturbation. We assume that the launch vehicle generates body rate commands slower than the torque commands are produced, which are then allocated and actuated by the aero-surfaces and/or engines of the launch vehicle. Upon this assumption, we subsequently divide the systems into two subsystems, one corresponding to slower dynamics and other corresponding to faster dynamics, and then design controllers for each one of them separately. Thus, the multipleloop sliding mode control design is composed of an outer loop, which generates body rate commands, and the inner loop which produces the torque commands. The single loop design takes two different approaches. The first is accomplished by generating the body rate that are calculating directly from the given in current time quaternion commands. The other approach is that of a higher-order quasi-continuous control using control T as the design variable. With the single loop control, time-scaling [11] no longer needs to be preserved. 4.1. Multiple loop sliding mode attitude control driven by sliding mode disturbance observer (SMC-SMDO) The smooth multiple-loop classical SMC-SMDO sliding mode controller [11] is designed in three steps: Step 1: The outer (guidance) loop the SMC-like smooth controller generates body rate commands (virtual control) _ xc ¼ 2K1 q ½qc þ K1 eq þ K2 s0 ) xc ¼ 2K-1 ½q_ c þ ðK1 þ K2 Þeq þ K2 K1 $c , q
where
Z
ð14Þ
t
eq dt,
s0 ¼ eq þK1
$ _ c ¼ eq
s0 2 R3 ,
eq ¼ qc q
ð15Þ
0
that provide the following outer loop (guidance) compensated dynamics (given the xc profile is tracked perfectly in the inner loop) e€ q þðK1 þ K2 Þ_e q þðK2 K1 Þeq ¼ 0,
eq ð0Þ ¼ eq0 ,
e_ q ð0Þ ¼ e_ q0
ð16Þ
Remark 1. The sliding variable s0 in (15) is driven to zero asymptotically by the SMC-like smooth virtual control xc in (14). The quaternion tracking error eq dynamics obey the second order linear homogeneous differential Eq. (16). It means that the output tracking error eq tends to zero as time increases upon a corresponding selection of the diagonal matrices K1 and K2 . Step 2: The inner (faster) loop smooth sliding mode controller driven by a sliding mode disturbance observer (SMC-SMDO) generates control torque commands T ¼ T1 þ T2
ð17Þ
404
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
where T1 compensates for the known RLV dynamics and provides for the desired sliding variable, so , stabilization dynamics as well as the desired overall body rate tracking error dynamics. This is _ c þJ0 K3 ex þ OJ0 x þ J0 K4 sx ) T1 ¼ J0 x _ c þOJ0 xc þ ðJ0 K3 OJ0 þ J0 K4 Þex þ J0 K4 K3 ge , T1 ¼ J0 x
g_ e ¼ ex
ð18Þ
and T2 is generated using a corresponding SMDO, which design will be presented in Step 3, to compensate for the perturbation term FðÞ in Eq. (3) ^ T2 ¼ J0 FðÞ ð19Þ The sliding variable, so , is designed Z t so ¼ eo þK3 eo dt, so 2 R3 , eo ¼ oc o
ð20Þ
0
and is driven to zero asymptotically via the SMC-SMDO (17)–(19). The following body rate compensated dynamics (given exact estimation of the perturbation term in (3) by ^ ¼ FðÞ and the T profile is allocated perfectly into commands to actuator SMDO, i.e. FðÞ deflections) is provided in the inner loop: e€ x þðK3 þK4 Þ_e x þK4 K3 ex ¼ 0,
ex ð0Þ ¼ ex0 ,
e_ x ð0Þ ¼ e_ x0
ð21Þ
Step 3: Sliding mode disturbance observer (SMDO) is designed for estimating the body rate dynamics in Eq. (3) that is rewritten in a form: x_ ¼ Uð Þþu
ð22Þ
-1 where Uð Þ ¼ J1 0 ðXJ0 xþFðÞÞ and u ¼ J0 T. Eq. (22) can be rewritten in a scalar format
o_ i ¼ FðÞi þ ui ,
8i ¼ 1,2,3
Auxiliary sliding variables, s~ i , are introduced ( s~ i ¼ oi zi z_ i ¼ ui vi
ð23Þ
ð24Þ
and are driven to zero in finite time by means of the classical SMC injection term vi ¼ ri signð~s i Þ,
ri 49Fi ðÞ9
ð25Þ
Filtering the injection terms (25) in the sliding modes s~ i ¼ 0 by a low pass filter (LPF) of a desired relative degree we obtain estimates of FðÞi ^ i ¼ LPF fvi g FðÞ
ð26Þ
Finally, the perturbation term FðÞ is reconstructed: ^ ¼ J0 FðÞ ^ þ OJ0 o FðÞ
ð27Þ
Remark 2. The sliding variable so in (20) is driven to zero asymptotically by the smooth SMC-SMDO control torque in Eqs. (18), (19), and (27). The body rate tracking error dynamics ex obey the second order linear homogeneous differential equation (21). It means that the output tracking error ex tends to become zero as time increases upon a corresponding selection of the diagonal matrices K3 and K4 . Writing inner and outer loop
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
405
tracking error equations in a damping ratio/natural frequency format 2 € _ xþ2nx n xþx nx ¼ 0
ð28Þ
it is easy to calculate the elements of the diagonal matrices K1, K2 , K3 and K4 to provide given damping factors (usually all are equal to 1.1 or so) and natural frequencies that provide for a sufficient time-scale separation between the control loops to Eqs. (16) and (21) for inner and outer loop compensated tracking dynamics. 4.2. Multiple loop Higher order sliding mode attitude control The outer loop HOSM controller generates body rate commands. The launch vehicle motion that is considered in the outer loop is described in quaternions in Eq. (1). The launch vehicle motion that is considered in the inner loop is described by Eq. (2). The control torque T is considered as a control that will be allocated into aerodynamic surface deflection commands, rocket engine chamber pressure commands, and reaction control system commands. Since the control action xc that is fed to an inner loop may be required to be smooth for actuator input, a third order outer loop is developed with a second order inner loop. The double loop continuous HOSM controller is designed in two steps. Step 1: For the outer loop, we let r0 ¼ qc q 2R3 , and we wish to drive r0 -0 in finite time. In order to achieve a sufficient smoothness of the body rate command xc ðtÞ, the outer € loop quasi-continuous HOSM control law is designed in terms of v2 ¼ x. 1 Denoting x ¼ 2Kq x, the sliding variable dynamics are derived 8 _ > < r_ 0 ¼ qc þx ð3Þ _ ¼ v1 x ð29Þ -rð3Þ 0 ¼ qc þ v 2 > : v_ 1 ¼ v2 Relative degree of the outer loop sliding variable r0 -0 is equal to 3. Therefore, the outer loop 3rd order quasi-continuous HOSM control law is designed xc ¼ 2K1 q x € ¼ v2 x v_ 2j ¼ a2j
s€ oj þ 2ð9s_ oj 9 þ 9soj 9
2=3 1=2
Þ
ðs_ oj þ 9soj 9
2=3 1=2
9s€ oj 9 þ 2ð9s_ oj 9 þ 9soj 9
Þ
2=3
9s_ oj þ 9soj 9
signðsoj ÞÞ
2=3
,
j ¼ 1,2,3
ð30Þ
signðsoj Þ9
By making a2j 40, j ¼ 1,2,3 large enough, the 3rd order HOSM control law control law (30) will drive r0 , r_ 0 , r€ 0 -0 in finite time. Step 2 For the inner loop: we let r1 ¼ xc x, and we wish to drive r1 -0 in finite time. In order to achieve sufficient smoothness of the torque command TðtÞ, the inner loop _ Denoting T ¼ quasi-continuous SOSM control law is designed in terms of v3 ¼ T. 1 1 J0 XJ0 x þ J0 T, the sliding variable r1 dynamics are derived ( _ c Fð ÞT r_ 1 ¼ x _ Þv3 € c Fð ð31Þ -r€ 1 ¼ x T_ ¼ v3
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
406
and the inner loop quasi-continuous 2-SMC control law is designed as T ¼ J0 TXJ0 x T_ ¼ v 3
v3j ¼ a3j
s_ 1j þ 9s1j 9
1=2
signðs1j Þ
9s_ 1j 9 þ 9s1j 9
1=2
j ¼ 1,2,3,
,
ð32Þ
By making a3j 40, j ¼ 1,2,3 large enough, the SOSM control law (27) will drive r1 , r_ 1 -0 in finite time. In order to implement the HOSM double loop controller (30) and (32) the sliding variable derivatives r_ 0 , r€ 0 , r_ 1 can be obtained via HOSM exact sliding mode differentiator (13). Remark 3. The designed double-loop HOSM controller is continuous and drives the quaternion tracking error to zero in finite time with its derivative. Therefore, full dynamical collapse is achieved via smooth control in systems with the bounded disturbance. In theory, we can continue creating arbitrary order sliding mode controls using this technique, which can achieve higher orders of accuracy. However, depending upon the application, higher orders of sliding mode control design may be difficult to implement. 4.3. Single loop Higher order sliding mode attitude control The same finite-convergence time quaternion tracking can be achieved by designing a single loop HOSM controller using the quasi-continuous higher order sliding mode control technique. The proposed single loop control structure is simpler than the double-loop one presented in Section 4.1. We let rs ¼ qc q, and we wish to drive rs -0 in finite time. In order to achieve the continuity of the torque command Tc ðtÞ, the single-loop quasi-continuous SOSM control _ Denoting T ¼ 1K ðJ1 XJ x þ J1 TÞ, the sliding law is designed in terms of v4 ¼ T. 0 0 2 q 0 variable rs dynamics are derived as 8 > < r€ s ¼ q€ þ 1 K_ q xþ 1 Kq Fð ÞþT 1_ 1_ 1 _ c ð3Þ 1 € 2 2 _ Þþv4 -rð3Þ K q Fð Þ þ Kq Fð s ¼ qc þ K q xþ K q xþ > 2 2 2 2 : T_ ¼ v 4
ð33Þ and the single-loop quasi-continuous SOSM control law is designed. This is T ¼ 2J0 K1 q TXJ0 x T_ ¼ v4 v_ 4j ¼ a4j
s€ sj þ 2ð9s_ sj 9 þ 9ssj 9
2=3 1=2
9s€ oj 9 þ 2ð9s_ sj 9 þ 9ssj 9
Þ
ðs_ sj þ 9ssj 9
2=3 1=2
Þ
2=3
9s_ sj þ 9ssj 9
signðssj ÞÞ
2=3
,
j ¼ 1,2,3
ð34Þ
signðssj Þ9
By making a4j 40, j ¼ 1,2,3 large enough, the control law (34) will drive rs , r_ s , r€ s -0 in finite time. That means achieving the exact tracking of the commanded quaternion profiles, or full dynamical collapse.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
407
Remark 4. It is worth noting that relative degree of system can be increased should we take into account the dynamics of the actuator. If the dynamics of the actuator are considered as parasitic ones, a limit cycle could occur and system’s stability must be re-evaluated [23].
5. Simulations A case study on the SLV-X Launch Vehicle studied under a joint DARPA/Air Force program called the Force Application and Launch from CONtinental United States (FALCON) program is presented. The flight control system of RSLV SLV-X based on single-loop and double-loop HOSM controller is simulated using Matlab/Simulink Version 6.5 with Aerospace Blockset. The launch vehicle model used is a high fidelity 6DOF model with table look up aerodynamic and torque coefficients and a realistic wind model. The model also simulates natural environments. The atmospheric model uses the Committee on Extension to the Standard Atmosphere (COESA) United States standard atmospheric values. The Gravity Model implements the geocentric equipotential ellipsoid of the World Geodetic System (WGS84). The Kennedy Space Center Wind Model and Dryden Wind Turbulence Model blocks implement the mathematical representation of wind disturbance. The simulation trajectories of the RSLV SLV-X are available up to 400 s. For testing of the controllers, 126 s of flight time was used, which corresponds to a staging event that occurs, and allows time to show recovery after that staging event. Winds were also used for selective plots. The wind disturbances are highest at the very beginning, and then tend to settle to zero at about 70 s into flight. Fig. 2 shows the wind angular rate disturbances (in degrees/s) acting on the vehicle. We see that peak winds reach about 10 1/s just after liftoff, and settle down at about 63 s. After about 105 s, which corresponds to about 120,000 ft altitude, wind disturbances are essentially zero. The results of the simulations using single and double-loop HOSM 12
Angular rate - q (degrees)
10 8 6 4 2 0 -2 -4 -6 -8 0
20
40
60
80
Time (sec) Fig. 2. Winds profile.
100
120
140
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
408
100 Pitch Cmd Pitch Sensed
90
Pitch (degrees)
80 70 60 50 40 30 20 0
20
40
60
80 Time (sec)
100
120
140
120
140
Fig. 3. Pitch angle tracking (HOSM single loop).
0.015
Pitch Error (degrees)
0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 0
20
40
60 80 Time (sec)
100
Fig. 4. Pitch tracking error (HOSM single loop).
controllers appeared to be very similar, and the presented plots are for the single loop controller only. Fig. 3 shows the pitch tracking (in degrees) via single-loop HOSM controller. One can see that the HOSM single loop tracks so accurate in the presence of the disturbance (Fig. 2) that it is hard to see a difference in the command vs. the sensed. The pitch tracking error is shown in Fig. 4. It is clear that on average the error, and also the corresponding sliding variable, is kept at zero. There are time instants that controller moves off the surface, but it recovers almost instantly. We see that the highest error magnitude is at 0.021.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
409
Pitch Error Derivative (degrees/sec)
0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 0
20
40
60 80 Time (sec)
100
120
140
120
140
Pitch Error 2nd Derivative (degrees/sec2)
Fig. 5. Pitch error derivative (HOSM single loop). 400 200 0 -200 -400 -600 -800 0
20
40
60 80 Time (sec)
100
Fig. 6. Pitch error 2nd derivative (HOSM single loop).
Fig. 5 shows the first time derivative of the pitch angle tracking error (that is proportional to the derivative of the corresponding sliding variable). We see that the single loop HOSM controller drives the first derivative of our variable to zero in finite time, and does very nicely in maintaining the derivative at zero in spite of the disturbances. The HOSM single loop controller also drives the 2nd time derivative of our sliding variable to zero in finite time as shown in Fig. 6. Figs. 7 and 8 show the effect of wind disturbances on pitch angular rate as well as the control torque. Fig. 9 shows a comparison of the pitch rate profiles between the traditional double loop SMC-SMDO [10,11] and HOSM single loop controller in the pitch channel. Wind disturbances were removed to more accurately show the differences in the angular rates.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
410
12 10
Pitch Rate (degrees/sec)
8 6 4 2 0 -2 -4 -6 -8 0
20
40
60 80 Time (sec)
100
120
140
120
140
Fig. 7. Pitch rate (HOSM single loop).
6
x 105
Pitch Torque (lbs ft)
4 2 0 -2 -4 -6 -8 0
20
40
60 80 Time (sec)
100
Fig. 8. Pitch control torque (HOSM single loop).
As we can see, even without the disturbances, the HOSM enforces pitch rate is much more aggressive. This is due to the fact that the use of the HOSM single loop algorithm foregoes controlling of the angular rates in order to provide more robustness. This is desirable only in situations where the launch vehicle is not carrying a human payload. There are angular rate restrictions on humans in space flight that would require angular rate control available in the double loop but not in a single loop. This would also be applicable to reusable launch vehicles in certain configurations. The single loop shown here is desirable only for expendable uncrewed launch vehicles such as the FALCON RSLV SLV-X.
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
411
Fig. 9. Comparison of body pitch rates: SMC-SMDO and HOSM Single Loop.
6. Conclusion The Higher Order Sliding Mode Attitude controls for launch vehicles for the multiple and single loop architectures are shown and tested on a simulation platform based on SLV-X Launch Vehicle. These controllers allow for customization of launch vehicle attitude controls depending on the type of vehicle used. Unmanned ELVs do not return to Earth and normally employ a high control authority. Furthermore, control authority can be fully utilized when the ELV is not human rated since the limits on angular rates for an unmanned flight are higher. Therefore the control approach for unmanned ELVs is to provide a simple, but robust, control algorithm like the single order HOSM that will utilize the full control authority of the launch vehicle. Since human rated launch vehicles require less aggressive angular rate profiles due to crew being onboard, a control approach that will minimize the angular rates but still provide adequate robustness is needed, in particular the SMC-SMDO double loop flight controller. References [1] John Hanson, Integration and Testing of Advanced Guidance and Control Technologies, In: Proceedings of the Advanced Guidance and Control Workshop, NASA/MSFC, July 2003. [2] C.E. Hall, A.S. Hodel, J.Y. Hung, Variable structure PID control to prevent integral windup, in: Proceedings of the IEEE 31st Southeastern Symposium on System Theory, 1999, pp. 169–173. [3] C.E. Hall, M.W. Gallaher, N.D. Hendrix, X-33 attitude control system design for ascent, transition, and entry flight regimes, in: Proceedings of AIAA Guidance, Navigation and Control Conference, AIAA-1998-4411, 1998. [4] J. Zhu, A. Scott Hodel, K. Funston, C.E. Hall, X-33 entry flight controller design by trajectory linearization—a singular perturbation approach, in: Proceedings of American Astronautical Society Guidance and Control Conference, Breckenridge, CO, January 2001, pp. 151–170.
412
J.E. Stott, Y.B. Shtessel / Journal of the Franklin Institute 349 (2012) 397–412
[5] A. Huizenga, T. Bevacqua, J. Fisher, D. Cooper, J. Zhu, Improved trajectory linearization flight controller for reusable launch vehicles, in: Proceedings of 42nd AIAA Aerospace Science Meeting, Reno, NV, AIAA 2004-0875, January 2004. [6] M. Johnson, T. Calise, E. Johnson, Evaluation of an adaptive method for launch vehicle flight control, in: Proceedings of AIAA Guidance, Navigation and Control Conference, Austin, TX, August 2003. [7] E. Johnson, T. Calise, H. El-Shirbiny, Feedback linearization with neural network augmentation applied to X-33 attitude control, in: Proceedings of AIAA Guidance, Navigation and Control Conference, 2000. [8] V. Utkin, J. Guldner, J. Shi, Sliding Modes in Electromechanical Systems, Taylor and Francis, London, 1999. [9] C. Edwards, S. Spurgeon, Sliding Mode Control, Taylor & Francis, Bristol, PA, 1998. [10] Y. Shtessel, J. Stott, J. Zhu, Time-varying sliding mode controller with sliding mode observer for reusable launch vehicles, in: Proceedings of AIAA Guidance, Navigation and Control Conference, Austin, Texas, August 2003. [11] Y. Shtessel, J. Stott, J. Zhu, Reusable launch vehicle control via time-varying multiple time scale sliding modes, in: Proceedings of the 17th IMACS World Congress Scientific Computation, Applied Mathematics and Simulation, Paris, France, July 2005. [12] A. Levant, Higher-order sliding modes, differentiation and output-feedback control, International Journal of Control 76 (9/10) (2003) 924–941. [13] A. Levant, Construction principles of 2-sliding mode design, Automatica 43 (4) (2007) 576–586. [14] A. Levant, Quasi-continuous high-order sliding-mode controllers, IEEE Transactions on Automatic Control 50 (11) (2005) 1812–1816. [15] L. Fridman, Y. Shtessel, C. Edwards, X. Yan, Higher-order sliding-mode observer for state estimation and input reconstruction in nonlinear systems, International Journal of Robust and Nonlinear Control 18 (2008) 399–412. [16] C. Hall, Y. Shtessel, Sliding mode disturbance observers-based control for a reusable launch vehicle, AIAA Journal on Guidance, Control, and Dynamics 29 (6) (2006) 1315–1329. [17] Y. Shtessel, C. Hall, M. Jackson, Reusable launch vehicle control in multiple time scale sliding modes, AIAA Journal on Guidance Control, and Dynamics 23 (6) (2000) 1013–1020. [18] U. Itkis, Control Systems of Variable Structure, Keter Publishing house, Jerusalem, 1976. [19] V.I. Utkin, Variable structure systems with sliding modes, IEEE Transactions on Automatic Control 22 (1977) 212–222. [20] S.V. Emelyanov, S.K. Korovin, L.V. Levantovsky, Higher order sliding regimes in the binary control systems, Soviet Physics, Doklady 31 (1986) 291–293. [21] A. Levant (L.V. Levantovsky), Sliding order and sliding accuracy in sliding mode control, International Journal of Control 58 (1993) 1247–1263. [22] D. Drake, M. Xin, S.N. Balakrishnan, A new nonlinear control technique for ascent phase of reusable launch vehicles, AIAA Journal of Guidance, Control and Dynamics 27 (6) (2004) 938–948. [23] I. Boiko, L. Fridman, A. Pisano, E. Usai, Analysis of chattering in systems with second-order sliding modes, IEEE Transactions on Automatic Control 52 (11) (2007) 2085–2101. [24] M.C. Pai, Design of adaptive sliding mode controller for robust tracking and model following, Journal of the Franklin Institute 347 (10) (2010) 1837–1849. [25] H. Yang, Y. Xia, P. Shi, Observer-based sliding mode control for a class of discrete systems via delta operator approach, Journal of the Franklin Institute 347 (7) (2010) 1199–1213. [26] I. Boiko, Frequency domain precision analysis and design of sliding mode observers, Journal of the Franklin Institute 347 (6) (2010) 899–909. [27] M. Basin, D. Calderon-Alvarez, Sliding mode regulator as solution to optimal control problem for non-linear polynomial systems, Journal of the Franklin Institute 347 (6) (2010) 910–922. [28] Y. Xia, Z. Zhu, C. Li, H. Yang, Q. Zhu, Robust adaptive sliding mode control for uncertain discrete-time systems with time delay, Journal of the Franklin Institute 347 (1) (2010) 339–357.