Application of Adaptive Backstepping for the Control of a Reusable

Launch Vehicle

Ashima. C. R

M.Tech Student

College of Engineering

Trivandrum

Dr. S. Ushakumari.

Professor

College of Engineering

Trivandrum

S. Geetha

DPD,GSLV

VSSC.

Trivandrum

2

Abstract

In this paper, a nonlinear Lyapunov based controller is used for the stabilization of the lateral dynamics and longitudinal

dynamics of a Reusable Launch Vehicle during its descent phase, inorder to cope up with the nonlinearities and uncertainties

of the sytem which are araised due to huge variation in Mach numbers and aerodynamic profiles, the control strategy based

on adaptive state feedback and backstepping control is employed. The novelty is in the use of a robust controller for high

speed launch vehicles that ensures the stabilization and the asymptotical convergence of the angle tracking error and the

estimation error. Simulation studies are conducted to ensure the effectiveness of the proposed scheme. From the simulation

results it is clear that the proposed controller works satisfactorily.

0.1. Introduction

The flight of a Reusable Launch Vehicle during its descent phase is subjected to a huge variation in Mach numbers and

adverse flight envelopes and the system must be stabilized in the midst of these uncertainties. In the literature, not many

non-linear control schemes have been proposed for the control of a Reusable Launch Vehicle. The inverted pendulum forms

a classical nonlinear and unstable system. Therefore the control of inverted pendulum [1] [2] has been studied to understand

the best control methodology for the control of a Reusable Launch Vehicle. In the recent years, the focus on methodologies

of control systems has shifted from linear to nonlinear systems. Though powerful linear design tools are available, nonlinear

design tools such as feedback linearization, gain scheduling, and backstepping can come up with accurate results retaining

useful nonlinearities.

The backstepping design procedure [3]is found to have a lot of advantages compared to feedback linearization and gain

scheduling techniques. But, the backstepping design methodology fails in the case of systems which cannot be converted to

either pure feedback forms or strict feedback forms and also in the presence of parameter uncertainties.

Benaskeur and Desbiens [1]proposed an adaptive two-loop cascade controller to overcome the fact that the system should

be written in a triangular or pure feedback form. The inner loop uses an adaptive backstepping control technique to regulate

the rod angle with regard for the cart motion and unknown rod length. Ebrahim et .al [2]proposed the same Adaptive

Backstepping Controller but treating every constant parameters in the system as unknowns such as mass of the pendulum,

1

mass of the cart, moment of inertia, gravitational constant and adaptation is given accordingly which would be a difficult task

for any linear control methods.

In [4], the autopilot design for reentry vehicle is carried out using an Adaptive Backstepping Control design technique. The

attitude equations are nonlinear and time dependent due to large variation of aerodynamic parameters and velocity profiles.

Apart from the traditional design methods which uses Euler angles as the state variables, here side slip angle and angle of

attack are selected as the state variables to avoid a non-minimum phase situation. Simulation studies conducted on a fully

nonlinear 6-DOF dynamic model produced satisfactory attitude control performance in conjunction with the optimization of

moment coefficients. Recently, researchers are also interested in Robust Adaptive Backstepping controller design methods [5]

which mitigates the problem of unmodelled dynamics, parameter drift and noise. Moreover the stability of a system may

be severly affected by some bounded disturbances and high rate of adaptation which can be overcome by incorporating a

continuous switching function in the parameter updation law.

In this paper, an Adaptive Backstepping Controller is used to stabilize the longitudinal and lateral dynamics of a Reusable

Launch Vehicle model by considering all the constant parameters in the system as unknowns.

This paper is organized as follows: section II deals with the modelling of a Reusable Launch Vehicle. Adaptive Back-

stepping Controller design is discussed in Section III and section IV gives a detailed description of the controller design for

the longitudinal dynamics of the system. Section V deals with the Adaptive Backstepping Controller design for the laretal

dynamics of the RLV which is followed by the simulation results and conclusions.

0.2. Dynamic model of a

Reusable Launch Vehicle

The system under consideration is a Reusable Launch Vehicle (RLV) during its re-entry phase [6]. During this time, the

aerodynamic forces become comparable to the gravitational forces. A conventional aircraft is usually equipped with three

control surfaces viz: a rudder, an elevator, and an aileron. But, the X-38 has only two sets of control surfaces: rudders and

elevons. The X-38 vehicle has a pair of rudders on top of each of the two vertical fins, a pair of elevon on lower rear of the

vehicle . Each surface can be controlled independently to obtain the required control action.

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Figure 1: Control surface deflections

The elevon angle for pitch control is given by averaging the elevon deflections.

e =eL + eR

2(1)

The aileron angle for roll control is given by taking the average of the difference of the elevon deflections.

a =eL eR

2(2)

Similarly, the total rudder angle for yaw control is given by taking the average of the rudder deflections.

r =rL + rR

2(3)

The non-linear set of equations of which describes the motion of the vehicle is as follows [6]:

p =1

Ixx((L + Lpp+ Lrr + Laa+ Lrr) (4)

q =1

Iyy(M+Mqq +Mee+Mrr) (5)

r =1

Izz(N +Npp+Nrr +Naa+Nrr) (6)

=ZVT

sin g sin VT

+

(ZqVT

+ 1

)q (7)

=Y

VT+Ypp

VT+

(YrVT 1)r +

g

VT (8)

3

In the above set of equations [4-8] p, q and r are the roll rate, pitch rate and yaw rate respectively. Here, is the angle of

attack, is the side slip angle, and is the flight path angle. The equations are represented in terms of aerodynamic forces

and moments, where L is called the rolling moment, M the pitching moment and N is the yawing moment. Ixx, Iyy, Izz are

the moment of inertia in the x, y and z directions respectively. is the roll angle, g is the acceleration due to gravity and Vt

is the vehicle velocity. The coefficients Y and Z in the equations represents side force and downward force respectively. The

dependence of the aerodynamic forces on the angle-of-attack and the side slip angle is crucial to stability and control.

0.3. Adaptive Backstepping Control

0.3.1. Basic Backstepping Technique

Backstepping designs by breaking down complex nonlinear systems into smaller subsystems, then designing control Lya-

punov functions and virtual controls for these subsystems and finally integrating these individual controllers into the actual

controller, by stepping back through the subsystems [3].

Consider a system of the form

x = f(x) + g(x)1

1 = f1(x, 1) + g1(x, 1)2

2 = f2(x, 1, 2) + g2(x, 1, 2)3

. (9)

.

.

k = fk(x, 1, ..., k) + gk(x, 1, ..., k)u

To show how to find a control Lyapunov function and a control law, a short design example is considered . The system

that is to be controlled is given below.

x = f(x) + g(x)

= a(x, ) + b(x, )u (10)

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where x Rn and R are state variables and u R is the control input. First is regarded as a control input for the

x-subsystem. can be chosen in any way to make the x-subsystem globally asymptotically stable. The choice is denoted

des(x) and is called a virtual control law. For the x-subsystem a control Lyapunov function, V1(x), can be chosen so that

with the virtual control law, the time derivative of Lyapunov function becomes negative definite.

V1(x) = V1xx = V1x(x)(f(x) + g(x)des(x)) < 0, x 6= 0 (11)

A new state is introduced which represents the error variable

= des(x) (12)

The system shown in equation (10) is then written in terms of these new variables

x =f(x) + g(x)( + des(x))

=a(x, + des(x)) + b(x, + des(x))u (13)

des(x)

x(f(x) + g(x)( + des(x)))

For the system given above a control Lyapunov function is constructed from V1(x) by adding a quadratic term which penalizes

the error variable ,

V2(x, ) = V1(x) +1

22 (14)

Differentiating V2(x, ) with respect to time

V2(x, ) =V1x(x)(f(x) + g(x)des(x) + g(x)

)+ (

a(x, + des(x)))+ {b(x, + des(x))u

des(x)

x(f(x) + g(x)( + des(x)))} (15)

Equation (15) can be rewritten in the following way if the variables that the functions depend on are omitted. To guarantee

stability V2 has to be negative definite. This can be achieved by choosing the control input, u in (15) as

u =1

b

(des(x)

x(f + g( + des(x))) a V1g k

)(16)

where k > 0. Then V2 becomes

V2 = V (f + gdes) k2 0 (17)

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If u is not the actual control input but a virtual control law consisting of state variables, then the system can be further

expanded by starting over again. Hence the backstepping design procedure is recursive.

0.3.2. Adaptive Backstepping Controller Design

The Backstepping controller design guarentees that by employing a static feedback, the closed loop state remains bounded

in the presence of uncertain bounded nonlinearities. While the Adaptive Backstepping Controller design employ a form of

nonlinear integral feedback and the underlying idea in the design of this dynamic part of feedback is parameter estimation.

The dynamic part of the controller is designed as a parameter update law with which the static part is continuously adapted

to new parameter estimates.

Adaptive Backstepping Controllers are dynamic and more complex than the static controllers. What is achieved with

this complexity is that, an Adaptive Backstepping Controller guarentees not only that the plant x, remains bounded, but

also regulation and tracking of a reference signal. In its basic form, the Adaptive Backstepping Control design employs

overparametrization and this means that the dynamic part of the controller is not of minimal order. Consider

x1 = x2 + (x1)

x2 = u

(18)

where is a known constant parameter and x2 as the first control input.Denote o as the estimated value for the parameter

and the estimation error e is given by

e = o (19)

Next the candidate Lyapunov function is selected as

V2(x, e) =1

2x21 +

1

22e (20)

where is the adaptation gain. with the control law

x2 = k1x1 (x1) = 1(x1, ) (21)

and the adaptation law

o = (x1)x1 (22)

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the derivative of the candidate Lyapunov function becomes negative defenite and is given by

V1 = k1x21 < 0 (23)

in the equation(21) 1 is called a stabilizing function for x2

The deviation of x2 from the stabilizing function is given by

z = x2 1(x1, ) (24)

Augmenting the Lyapunov function by adding the error variable

V2(x, z, e) = V1(x1, e) +1

2z2 (25)

By the proper selection of u, the overall Lyapunov function V2, becomes negative defenite which implies that as x1 tends

to zero, then z also tends to zero asymptotically.

0.4. Adaptive Backstepping Controller design

for Longitudinal Dynamics of RLV

In longitudinal dynamics, angle of attack () and pitch rate (q) equations are considered for the design of backstepping

controller. The system considered after taking the assumption that the effect of coupling is negligible is given as

= Z sinVt + q

q = 1Iyy (M sin+Mqq +Me sin e)

(26)

The state variables are selected as x1 = ;x2 = q. The control variable is u = e. The system of equations can now be

expressed as

x1 =ZVtsinx1 + x2

x2 =1Iyy

[Msinx1 +Mqx2 +Mesinu]

(27)

The control law is to be designed such that the system stabilizes for whatever be the initial conditions. For applying the

Adaptive Backstepping Control design procedure, the system can now be expressed as

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x1 = 1sinx1 + x2

x2 = 2sinx1 + 3x2 + 4sinu

(28)

where 1, 2, 4 are the unknown parameters in the system.

The first error variable is defined as

e = x1 sp (29)

where sp is the desired set point. Using the Lyapunov function

V1 =1

2e2 (30)

and using the derivative of the Lyapunov function, the virtual control law can be formulated as

xdes2 = 1sinx1 + sp k1e (31)

where k1 > 0 and is a design parameter which guarentees V1 < 0. The second error variable is defined as

= x2 xdes2 (32)

By augmenting the Lyapunov function V1 with the error variable and the unknown parameters in the system, we get

V2 =1

2e2 +

1

22 +

1

2121e +

1

2222e +

1

2323e +

1

2424e (33)

where 1e, 2e, 3e, 4e are the parameter estimation errors of 1, 2, 3, 4 where: e = o and 1o, 2o, 3o, 4o

are the parameter estimates with 1, 2, 3, 4 are the adaptation gain constants. With the control law

The first derivative of the Augmented Lyapunov function becomes

8

V2 = ee+ +1

11o1o +

1

22o2o +

1

33o3o+

1

44o4o

V2 = [2sinx1 + 3x2 + 4sinu+ 1cosx1 + k1e sp]

+ee+1

11o1o +

1

22o2o +

1

33o3o +

1

44o4o

(34)

where 1e, 2e, 3e and 4e are the parameter estimation errors of 1, 2, 3 and 4 where e = o and stands

for 1, 2, 3, 4. The variables 1o, 2o, 3o, and 4o are the parameter estimates with 1, 2, 3, and4 as the adaptation gain

constants. With the control law

udes = 14o sin1{

2osinx1 3ox2 1ocosx1 k1e sp k2} (35)

and the parameter update laws given by

1o = 1cosx1

2o = 2sinx1

3o = 3x2

4o = 4sinu

(36)

The derivative of the augmented Lyapunov function becomes negative defenite

V2 = k1e2 k22 0 (37)

where k1 > 0, k2 > 0. Therefore by Laselles theorem, the system is globally asymptotically stable at the equilibrium point

of the system.

0.5. Adaptive Backstepping Controller design for Lateral Dynamics of RLV

The side slip angle and the yaw rate r are used to completely define the lateral dynamics of the system. The equations

of motion for the lateral dynamics are as follows.

=YVtsin r

r =1

Iz[Nsin +Nrsinr]

(38)

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For the ABC design procedures, the state variables are selected as x1 = ;x2 = r and control variable is selected as u = r.

The resulting mathematical model of the lateral dynamics of RLV can be written as,

x1 =YVtsinx1 x2

x2 =1

Iz[Nsinx1 +Nrsinu]

(39)

The system can be expressed in terms of unknown but constant parameters as

x1 = 1sinx1 x2

x2 = 2sinx1 + 3sinu

(40)

The first error varible is

e = x1 sp (41)

The candidate Lyapunov function for the system is

V1 =1

2e2 (42)

The virtual control law to make the Lyapunov function V1 to make negative defenite is given by

xdes2 = k1e+ 1sinx1 sp (43)

The second error variable is chosen as

= x2 xdes2 (44)

With which the augmented Lyapunov function becomes

V2 =1

2e2 +

1

22 +

1

2121e +

1

2222e +

1

2323e (45)

where 1e, 2e, 3e are the parameter estimation errors of 1, 2, 3 where e = o and 1o, 2o, 3o, are the

parameter estimates with 1, 2, 3, are the adaptation gain constants.

The first derivative of the augmented Lyapunov function can be made negative defenite by the application of the control

law

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udes =1

3osin1[3osinx1 sp + 1ocosx1 + k1e k2] (46)

where k1 > 0, k2 > 0 along with the parameter update laws as given by

1o = 1cosx1

2o = 2sinx1

3o = 3sinu

(47)

with [44] and [45], the derivative of the augmented Lyapunov function becomes

V2 = k1e2 k22 (48)

making the system [38] globally asymptotically stable.

0.6. Simulation results and Discussions

Figure 2: Angle of attack Vs time

Figure 2 shows the variation of angle of attack with time in seconds for an initial value of 20 degrees and for different

values of k1 and k2 and figure 3 shows the variation of pitch rate for the same initial condition. The plant parameters are

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Figure 3: Pitch rate Vs time

Figure 4: Perturbation Studies on Longitudinal Dynamics

perturbed by 50 and 150 percentage and the results are given in Figure 4. This clearly indicates that the longitudinal dynamics

of the system is stabilized by the controller.

Figure 5 shows the variation of side slip angle with time in seconds for an initial value of 20 degrees and different values

of k1 and k2. Figure 6 shows the variation of yaw rate with respect to k1 and k2. Figure 7 shows the rate of change of side slip

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Figure 5: Side slip angle Vs time

Figure 6: Yaw rate Vs time

angle for the same initial conditions. From the simulation results it is clear that the lateral dynamics of the Reusable Launch

Vehicle is also stabilized.

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Figure 7: Rate of change of Side slip angle

0.7. Conclusions

The Adaptive Backstepping Controller design procedure has been implemented for the stabilization of longitudinal and

lateral dynamics of a X-38 Reusable Launch Vehicle. Apart from the Backstepping design procedure in which only non-

linearities had been taken care of, in the ABC design uncertainties associated with the constant parameters of the system is also

dealt with. Simulation results verify the fact that in the midst of non-linearities and uncertainties, the system asymptotically

converges to zero.

The relatively large estimation time and over-parametrization are the two disadvantages of this control scheme when

compared with the Backstepping design procedure. This can be reduced either by constrained adaptive backstepping method

or by incorporating a switching function with the parameter update law, which will be investigated as a part of the future

work.

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