American Institute of Aeronautics and Astronautics1

Nonlinear Stochastic Control Part II: Ascent Phase Control of Reusable Launch Vehicles

Yunjun Xu1

Department of Mechanical, Materials, and Aerospace EngineeringUniversity of Central Florida, Orlando, FL, 32816

Ming Xin2

Department of Aerospace Engineering, Mississippi State University, Starkville, MS 39762

Prakash Vedula3

School of Aerospace and Mechanical Engineering, University of Oklahoma, Norman, OK 73019

In designing a robust ascent phase control for reusable launch vehicles, uncertainties such as variations in aerodynamics, jet effects, hinge moments, mass property, and navigation processing, etc. have to be considered and normally time and labor intensive Monte Carlo simulations are used in order to achieve a desired tracking performance distribution. In this paper, a systematic stochastic control design method based upon a direct quadrature method of moments proposed in Part I [26] will be applied to the ascending phase attitude control problem. In conjunction with a nonlinear robust control and an offline optimization through nonlinear programming, any order of stationary statistical moments can be directly controlled. Two simulation scenarios of the X-33 ascentphase control have been used to demonstrate the capabilities of the proposed method and the results are validated by Monte Carlo runs.

I. Introductionisturbance, random noise, uncertainty parameters, and approximation in mathematical modeling are inherent to most of dynamical systems. For example, aerodynamic forces acting on aerial vehicles cannot be modeled

exactly and normally its variation can be regarded as a random noise. Stochastic control is one of the corresponding technical approaches aiming to control the effects of these random deviations such that desired statistical characteristics can be achieved. However, control of a stochastic dynamical system is a more challenging problem as compared to its deterministic counterpart.

Increasing safety and reducing costs when placing payloads into Earth orbits have been a driving force in reusable launch vehicle (RLV) research, and flight control is one of the key enabling components to accomplish this goal. The ascent phase control involves attitude maneuvering through a wide range of flight conditions such as wind disturbances, plant uncertainties, engine failure, and aero-surface locks etc. The conventional PID control design is carried out by linearizing the system at a series of operating points and the gain scheduling technique is used. The significant drawback of this design is that the gain table will become prohibitively large if a wide range of possible mission, payload, and anticipated failure modes has to be covered and there are always cases that cannot be considered in the design stage. This results in a long and expensive vehicle design cycle. Several nonlinear control techniques have been proposed in the past decade to achieve the adaptive and robust capabilities. Such techniques include dynamic inversion [1], sliding mode control [2], optimal control based Theta-d method [3], neural adaptive control [4], and trajectory linearization [5], to name a few. Although these designs exhibited good adaptive and robust results, the tracking performance was limited to deterministic ones. In order to consider possible variations in the system and study the statistic performance of the controlled system, Monte Carlo type simulations have been used in conjunction with the above mentioned deterministic control methods [6]. This approach generally has a

1 Assistant Professor, The Department of Mechanical, Materials, and Aerospace Engineering, University of Central Florida,

Email: yunjunxu@mail.ucf.edu, AIAA Senior Member.2 Assistant Professor, The Department of Aerospace Engineering, Mississippi State University, Email: xin@ae.msstate.edu,

AIAA Senior Member.3 Assistant Professor, the School of Aerospace and Mechanical Engineering, University of Oklahoma.

D

AIAA Guidance, Navigation, and Control Conference10 - 13 August 2009, Chicago, Illinois

AIAA 2009-5956

Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

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large computational load and the design process is labor intensive. Therefore, a systematic stochastic control design would be more efficient.

Systematic stochastic control methods have been investigated extensively and currently the methodologies for linear systems have been well studied [7-11]. However, controlling nonlinear stochastic systems is a far more challenging task. In brief, most of the current methods [12-16] in solving nonlinear stochastic control problems have one or several of the following limitations: (a) the difficulty in solving the Fokker-Planck equation (FPE), (b) some expectations of nonlinear terms cannot be obtained effectively, (c) the form of the response PDF has to be known a priori, (d) it is not easy to handle inequality and equality constraints, and (e) stability is not guaranteed. The focus of the paper is on applying an innovative approach to solve the associated FPE in the corresponding stochastic system. This approach belongs to the category (somewhat promising) of approximation methods [12-16], through which the pre-specified response mean and covariance in the steady state can be achieved.

This approach, which was proposed in the Part I [26, 28], involves representing the state PDF of the closed-loop system in terms of a finite summation of Dirac delta functions, whose weights and locations (called “abscissas”) are determined based on moments constraints. Using a small number of scalars, the method is able to efficiently and accurately model stochastic processes described by the multidimensional FPE through a set of ordinary differential equations (ODEs). If the additive noise is regarded as a bounded uncertainty in the state function, the asymptotical stability can be guaranteed by the proposed nonlinear robust control design (deterministic). If the additive noise is considered separately, the stability is guaranteed up to the highest order of the moments constrained in the stochastic control design (stochastic).

This paper is organized as follows. First, we describe the dynamics model, uncertainty, noise, and control objectives for the ascent phase control of the X-33 RLV. In Section III, a detailed stochastic control design will be described. In this section, the overall control algorithm will be illustrated in Section III.A; a modified nonlinear robust control will be demonstrated in Section III.B; the direct quadrature based method will be proposed in details in Section III.C; in Section III.D, a nonlinear programming technique will be used to find the optimum control parameters offline. After that two simulation scenarios will be used to demonstrate the capabilities of the proposed control methodology: a step input response and a high-fidelity X-33 ascent profile tracking control. Conclusions are discussed in Section V.

II. Problem DefinitionThe ascent phase of a RLV follows a preset path from launch to main engine cutoff. The flight control of the

ascent phase is primarily to control the body attitude to track commanded Euler angles that are computed from the ascent guidance law [17].

A. RLV Attitude Dynamics

The dynamics equation that governs the rotational motion of the rigid body RLV is given by

1 1

( )

+ +

S

J J J v(1)

where the state vector T includes the roll, pitch, and yaw Euler angles, respectively; is the angular

velocity; is the skew symmetric matrix of the angular velocity, and ( )S is defined as

1 tan sin tan cos

( ) 0 cos sin

0 sin / cos cos / cos

S (2)

The torque in Eq. (1) includes contributions from the control effectors only. Wing body aerodynamic contributions to vehicle moments are not included in the model of this study because these effects are proportional to the body-axis rates, which are typically small during the ascent phase [3].

In Eq. (1), J is the time-varying moment of inertia, and its uncertainty can be modeled as 1ˆ J I + J , where

the symbol with “ ” denotes the nominal value. Furthermore, the uncertainty associated with the torque is assumed to be 2 ˆ . Both uncertainties are bounded by 5% . Here I is the identity matrix with proper

dimensions.The additive noise v , which may come from environmental perturbations and launch vibration, is characterized

as a Gaussian noise with a zero mean and a covariance matrix of Q . To track the commanded attitude profile,

( ) y h x (3)

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is used as the output of the system. The relative degree of the system is two.

B. Control Objectives

The objective of the controller design is to track the command profile of the Euler angles during the X-33 ascent phase, while achieving desired (attainable) steady state statistic characteristics, such as the standard deviation or any high order moments.

III. Nonlinear Stochastic Control Design

A. Control Algorithm

The proposed control algorithm is described step by step here.Step 1: A nonlinear robust controller will be developed such that the closed-loop system can track the desired

command profile asymptotically and is robust with respected to the bounded functional (in the state function) and parametric uncertainties (in the input matrix) as described in Section III. B. In this step, the additive noise v will not be taken into account. Note that the control parameters are unknown but constrained by the asymptotical stability requirement. Baseline values of the control parameters will be designed using the subsequent optimization based on steady state statistic requirement. The key point in designing this controller is to avoid any discontinuous functions as preferred by the associated Fokker Planck equation (as described in Section III. C).

Step 2: Once the nonlinear controller formulation is available based on Step 1, the additive noise will be considered and the closed-loop dynamics model will be converted into the Itô stochastic differential equation (SDE). If the process described by the SDE is a Markovian diffusion process, the PDF characterizing this process is then governed by the FPE [18].

Step 3: As described in Section III. C, through the proposed direct quadrature method of moments [24, 26, 28], the PDF that is governed by the FPE (a partial differential equation) can be related by a set of ordinary differential equation (ODE).

Step 4: As described in Section III.D, a nonlinear programming algorithm will be used to optimize the control parameters designed in Step 1 by minimizing the difference between the desired and actual steady state statistical moments governed by the associated FPE.

Step 5: Up to now, the control design is accomplished and the system will be tested in Monte Carlo simulations for the purpose of validation.

Also note that Step 1 through Step 5 are offline steps through which proper control parameters can be found such that the closed-loop system will have a desired tracking performance and preferred stochastic characteristics.

B. Step 1: Nonlinear Control Design

For the convenience, the dynamics model of Eq. (1) without considering the additive noise is rewritten here as

1( ) x f x B (4)

in which ]T T T x is the state, and the state function 1( )f x is

1 1

( )

Sf

J J(5)

The nominal model used in the controller design include the state equation

1̂ˆ ˆ( ) x f x B (6)

and the nominal output model ˆ y h is the same as Eq. (3). The nominal input matrix B̂ is described by

1ˆ

ˆ

0B =

J(7)

whereas the actual input matrix B is

1

1 2ˆ

0

B =

I + J(8)

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Here a new nonlinear robust control method is used where no discontinuous functions (different from most of the sliding mode control methodologies [19-22]) are involved because of the continuity requirement in the associated Fokker-Planck equation as described in Section III.C. Based on [23], the nonlinear robust controller is derived as

1 1

12

ˆ ˆ ˆ0 0ˆ ˆˆ( , ) [ ]dL L L

B f fx, h y h+ e + s (9)

in which the control parameters 0 0 , , and are found from

1

2ˆ 0

ˆ( ) d L f

I D s = F + D y h+ e s (10)

in order to satisfy the asymptotical tracking stability. The sliding surface is 0 � s e e , where d e and

d e respectively. “L” and the subscript “d” denote the Lie derivative operator and desired values

respectively. The asymptotic tracking stability proof can be found in [23]. Also note that “ ” is the dot product operation.

F and D in Eq. (10) are state function uncertainty and input matrix uncertainty respectively. It is not easy if not

possible to derive the uncertainty bound for the state function 1,2,3i iF

F where 2 2

ˆ ˆˆ

i i iF L h L h f f

. Therefore, in

the simulation, values after trial and error are applied. The uncertainty bound D is derived next.

First, the uncertainty bound for the input matrix D is , 3ij ijD i j where ij is calculated by

1 1

1

ˆ ˆˆ( ) ( )L L L L

B f B f

I + h x h x (11)

in which

1̂

1 2ˆ ˆ ˆ ˆ/ [ , ]L

fh x A A A , 3 3

1ˆ A , 3 3

2ˆ A (12)

Also, since the nominal values of the first three functions in 1̂f are the same as those of 1f , the uncertainty

bounds can be further simplified as

1 1

1

1ˆ

11

2 1 2 2

ˆˆ ˆˆ ˆ

ˆ ˆˆ ˆ

LL

f fhh

I + B B AB ABx x

A I + J JA

(13)

Eq. (13) can be further simplified as

11 12 1 2 2

ˆ ˆˆ ˆ A J I + JA I (14)

Note that in computing the bounds ijD , the sign of each entry in the matrix needs to be taken into account.

C. Steps 2 and 3: Fokker Planck Equation and Direct Quadrature Formulation

Once the nonlinear robust controller is designed, the closed-loop system can be written considering the additive noise as

1 2 0( ) ( , , ) �x f x B Gv f x Gv (15)

in which 2 2, 1,...,6[ ]i if f and 2,if is the thi component of 2f ; the diffusion matrix G is

3 30T

G = I (16)

and 3 3I is a 3x3 identity matrix.

The 1st order ODE Eq. (15) can be converted into the Itô form as

2 0( , , ) ( )d dt d t x f x G (17)

According to Jazwinski [18], ( ) ~ ( ) /t d t dtv and ( ) (0, )d t dt� � is a normal distribution.

As proven in [18] the process described by the SDE (Eq. 17) is a Markovian diffusion process. The PDF characterizing this process is governed by the FPE as

26 6 6

2,

1 1 1

1

2

T

iji

i i ji i j

ppfp

t x x x

GQG(18)

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where ( )p p x is the state PDF.

Here, a new quadrature based moment approach is proposed for solving the FPE efficiently. This method has been used to address the population balance problems in multiphase flows [24] and nonlinear filtering problems [25]. In this paper, the method is extended to the nonlinear stochastic control of the X-33 in ascent phase. It involves the approximation of the state PDF in terms of a finite summation of Dirac delta functions as

6

1 1

( ) [ ]N

j jj

p t w t x x

x (19)

where N is the number of nodes, ( )w w t denotes the corresponding weight for the node , 1,..., N .

( ), 1,...,6j jx x t j represents the property vector of node , called “abscissas” here. The weights and

abscissas are determined based on the constraints from the evolution of moments. Using a small number of scalars (in the Dirac delta function), this method is able to efficiently and accurately model stochastic processes described by the multivariable FPE through a set of ordinary differential equations (ODEs) as demonstrated below.

First, let us substitute Eq. (19) into the LHS of the Eq. (18), and the governing equation of the weights and abscissas can be derived as

6 66

1 1 11 1,

6 66'

1 1 11 1,

[ ] [ ]N N

jjj j l l

jjj l l j

N Nj

j l jjj l l j

xwpx x w x x

t t tx

xww

t t

(20)

where [ ]j j jx x � and ' /j j jx

� .

If the weighted abscissas j jw x � is introduced, after some manipulations, Eq. (20) can be written as

6 6 66 6' '

1 1 1 11 1, 1,

N Nj

j j l j l jj jj l l j l l j

w wpx

t t t t

(21)

Notice that w , j , and j are functions of time, thus the partial derivatives of the functions can be written as

total derivatives.6 6 66 6

' '

1 1 1 11 1, 1,

N Nj

j j l j l jj jj l l j l l j

ddw dwpx

t dt dt dt

(22)

Using the definitions/ , 1,...,dw dt a N � (23)

and / , 1,...,6; 1,...,j jd dt b j N � (24)

Equation (22) can be derived as6 6 66 6

' '

1 1 1 11 1, 1,

N N

j j l j l j j

j jj l l j l l j

px a b

t

(25)

and the FPE (Eq. 18) can be written in terms of the multi-variable Dirac delta function as6 6 66 6

' '

1 1 1 1 11 1, 1,

( )N N N

j j l j l j j

j jj l l j l l j

a x a b S

x x (26)

where the RHS of Eq. (26) is the RHS of Eq. (18), i.e.6 6 6 2

2,

1 1 1

[1 / 2 ( ) ]( )

Tiji

i i ji i j

ppfS

x x x

xGQG

x (27)

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There are total (1 6) 7N N parameters which need to be found in order to construct the state PDF p tx :

a and , 1,...,6, 1,...,jb j N . The direct quadrature method applies an independent set of user defined

moment constraints to construct 7N algebraic ordinary differential equations (ODEs).The 1 2 6, ,...,k k k moments of Eq. (26) can be written as follows

6 61 1

61

6 6 6 66'

1 6 1 61 1 11 1 1, 1

6 66'

1 61 1 1, 1

... ...

...

N Nk kk k

j l j l j l

jj l l l j l

Nkk

l j j l

j l l j l

x x a dx x x x a dx

x x b dx

61

6

1 61

... ( )kkl

l

x x S dx

x x

(28)

The first term in the LHS of Eq. (28) can be simplified as

61

6 6

1 61 61 11 1

... ... jN N

kkkj j

j j

x x a dx dx x a

(29)

whereas the second term in the LHS of Eq. (28) is6 6 66 6

'1

1 1 1 11 1, 1

... ... lm

s

N Nkk

m j l j N j lj jm l l j l

x x a dx dx k x a

(30)

In the same way, the third term of the LHS in Eq. (28) is

61

6 66 61'

11 61 1 1 11, 1,

... j l

s

N Nk kkk

j j l N j j l jj jl l j l l j

b x x dx dx k x x b

(31)

Therefore through Eqs. (29)-(31), the 7N unknown parameters can be found in

1 6

6 66 61

,...1 1 1 11 1,

1jl l

N Nkk k

j l j j l j k kj jl l l j

k x a k x x b S

(32)

where the RHS of Eq. (32) can be written as 1 6 1 6 1 6

1 2,... ,... ,...k k k k k kS S S� . According to the theory proposed in Part I of

the paper [26, 28],

61

1 6

61 11

61

,... 1 61 61

61

1 1 1 1 61 1

...

( ) ( ,... )i i i

s

kk ik k

ii

Nkk k kk

i i i i N ii

pfS x x dx dx

x

k w t x x x x x f x x

(33)

when i j ,

61

1 6

1 6

26 62

,... 1 61 61 1

66 6

,...1 1 1 1

[ (1/ 2 )1...

2

/ [1/ 2 ]l

Tijkk

k ki ji j

Nk T

i j l i j ijx x

i j l

pS x x dx dx

x x

w k k x x x

GQG

GQG

(34)

whereas when i j ,

61

1 6

1 6

22

,... 1 61 6 2

62

,...1 1

[1/ 2 ]...

( 1) / [1/ 2 ]l

Tkk ii

k ki

Nk T

i i l i ijx x

l

fS x x dx dx

x

w k k x x

GQG

GQG

(35)

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Thus, the 7N ODEs can be constructed using a set of independent moment constraints 1 6,...,k k as shown in Eq.

(32). In the steady state case, for any selected nonnegative integer 1 2 6, ,...,k k k , the corresponding stationary

moments of the PDF is governed by

1 6,... 0k kS (36)

D. Step 4: Nonlinear Programming

In sections III.B and III.C, the closed-loop system and the formulations of the approximation solution of its stochastic differential equation have been given. Here the unknown control parameters 0 and will be found

through an offline constrained nonlinear programming solver. The goal is to select control parameters such that the steady state value of the statistic characteristics can be controlled to the desired values.

The performance index to be minimized is the 2-norm of the error between the desired and actual stationary

1 2 6, ,...,k k k moments as

1 2 61 2 6 ......

2

k k kk k kdJ M M (37)

in which the performance index is a function of abscissas, weights, and control parameters. The inequality constraints comprise of the stability requirements 0η > and 0 0> [23], and property of the PDF 0w .

1

1N

w

(property of the PDF) and Eq. (36) are equality constraints.

It is well-known that the convergence of NLP typically is not guaranteed. However, scaling the system modeling and a proper initialization (in terms of abscissas, weight and control parameters) in general will help.

As a brief conclusion of the algorithm, (1) the closed-loop tracking system is asymptotically stable as described in Step 1; (2) any higher order steady state moments of the state/output variables can be controlled if this particular moment is included in Eq. (37); (3) the system will be stable up to the highest moments included in Eq. (37) because this moments are controlled; (4) during the derivation, the state process has not been assumed to be Gaussian; (5) the control parameters can be found through this design process and no Monte Carlo runs are needed.

IV. Simulation ResultsThe controller proposed in the paper will be tested in two simulation scenarios and validated through 10,000

Monte Carlo runs: a step input and a command profile obtained from a high-fidelity simulation of the X-33 RLV [3]. Here, the step size of the simulation is set to be 0.05t s , the Euler-Maruyama scheme [27] is used in the Wiener

integral, the torque command is saturated at 610 N m , and the uncertainty between the torque command and the actual torque applied is bounded by 5% . The uncertainty of the time-varying inertia is assumed to be 5%around the nominal value defined as

11 13

22

13 33

0

0 0

0

I I

I

I I

I (38)

The additive noise associated with the dynamics model is assumed to be2 2([25,25,25])(deg/ )diag sQ (39)

The goals of the controller are 1) able to track the preset trajectory profile, and 2) able to control the distribution of the steady state performance in terms of mean and standard deviation. In the second goal, the desired mean equals to the desired Euler angle, and the desired standard deviations are 0.2o

, 0.2o , and 0.2o

. In

the simulation result, “SC” is used to denote the performance achieved using the proposed stochastic design and validated by Monte Carlo runs (Step 5 in the algorithm), “NRC” denotes the robust nonlinear tracking control in which the baseline control parameters 0 [1,1,1]T and [1,1,1]T are used.

The moment constraints used in the nonlinear constrained optimization are selected to be

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1

2

3

[1,0,0,0,0,0,2,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,2,2,2,2,2,0,0,0]

[0,1,0,0,0,0,0,2,0,0,0,0,1,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,0,0,0,0,2,2,2]

[0,0,1,0,0,0,0,0,2,0,0,0,0,1,0,0,0,1,0,0,0,1,1,1,

k

k

k

4

5

6

0,0,0,0,1,0,0,0,1,0,0]

[0,0,0,1,0,0,0,0,0,2,0,0,0,0,1,0,0,0,1,0,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0]

[0,0,0,0,1,0,0,0,0,0,2,0,0,0,0,1,0,0,0,1,0,0,1,0,1,0,1,0,0,0,1,0,0,0,1]

[0,0,0,0,0,1,0,0,0,0,0,2,0,

k

k

k

0,0,0,1,0,0,0,1,0,0,1,0,1,1,0,0,0,0,1,0,0,0]

(40)

for the case where the number of node 5N after trials and errors is used. In this moment description (Eq. 40), for

example, 1 0 01 2 6 1 6... ( ) ...x x x p dx dx

x is represented by the first column [1,0,0,0,0,0]T , which gives the

mean value of the roll angle . As another example, 2 0 01 2 6 1 6... ( ) ...x x x p dx dx

x is represented by the

seventh column [2,0,0,0,0,0]T , which can be used to calculate the autovariance of the roll angle .

A. Case 1 – Step Response

First, the controller is tested in the case of tracking an ad-hoc step command 10od , 25o

d , and 10od

from the initial condition of 0 80o , 5oo , and 0 50o . One of the control parameters is set to be

0 [1,1,1]T . It takes roughly 177.48 seconds for the nonlinear programming solver to obtain the other set of

parameters [3.30,3.03,3.25]T . Note that (1) the computational cost is much less than the cases if the typical

iterative Monte Carlo simulations are used; (2) It would also work if both 0 and are regarded as the parameters

to be optimized.The mean values of the Euler angles are shown in Figs 1(a)-1(c), and the applied torques for the proposed

method are plotted in Figs. 1(d)-1(f). When Figs. 1(a)-1(c) (or the mean values as shown in Table 1) are examined, it can be seen that both controllers (“SC” and “NRC”) can track the command successfully. It is shown in Figs. 1(d)-1(f) that relatively larger (still under the saturation level) torques are applied initially for the proposed stochastic control because of the larger control parameter .

0 2 4 6 8 10-20

0

20

40

60

80

Time (s)

Roll

Angle

(deg)

NRC

SC

Command

Fig. 1(a) Mean value of the roll angle history in response to the step input

0 2 4 6 8 10-5

0

5

10

15

20

25

30

Time (s)

Pitc

h A

ngle

(d

egre

e)

Command

SCNRC

Fig. 1(b) Mean value of the pitch angle history in response to the step input

0 2 4 6 8 1010

20

30

40

50

Time (s)

Yaw

Ang

le (

degr

ee)

CommandSC NRC

Fig. 1(c) Mean value of the yaw angle history in

0 2 4 6 8 10-1

-0.5

0

0.5

1x 10

6

Time (s)

Rol

l Tor

que

(ft-

lb)

NRC

SC

Fig. 1(d) Mean value of the torque in the roll

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response to the step input direction

0 2 4 6 8 10-1

-0.5

0

0.5

1x 10

6

Time (s)

Pitc

h T

orqu

e (f

t-lb

)

NRC

SC

Fig. 1(e) Mean value of the torque in the pitch direction

0 2 4 6 8 10-1

-0.5

0

0.5

1x 10

6

Time (s)

Yaw

Tor

que

(ft-

lb)

NRC

SC

Fig. 1(f) Mean value of the torque in the yaw direction

The advantage of the proposed stochastic control method can be seen from Figs. 1(g)-1(i) as well as Table 1, where the standard deviations of the tracking performance characteristic are shown. Note that “Opt” in Table 1 denote the results obtained in the offline nonlinear optimization (Step 4 in the algorithm). First the standard deviations achieved using the proposed method equal to the desired value of 0.2o , while the values obtained through “NRC” do not. Second, the values validated through Monte Carlo runs match well with what have been obtained in NLP.

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Sta

ndar

d D

evia

tion

Rol

l Ang

le (d

egre

e)

SCNRC

Command

Fig. 1(g) Standard deviation of the roll angle in response to the step input

0 2 4 6 8 100

0.2

0.4

0.6

0.8

Time (s)

Sta

ndar

d D

evia

tion

Pitc

h A

ngle

(deg

ree)

Command

SC

NRC

Fig. 1(h) Standard deviation of the pitch angle in response to the step input

0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (s)

Sta

ndar

d D

evia

tion

Yaw

Ang

le (

degr

ee)

Command

SC

NRC

Fig. 1(i) Standard deviation of the yaw angle in response to the step input

Table 1 Step ResponseCase 1 Desired NRC Opt SC

-10o -9.8o -10o -9.98o

25o 25.02o 25o 25o

10o 10.1o 10o 10.01o

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0.2o 0.535o 0.2o 0.201o

0.2o 0.495o 0.2o 0.198o

0.2o 0.54o 0.2o 0.205o

B. Case 2 – X-33 Ascent Phase Control

The main objective of the simulation in this section is to demonstrate the capability of the proposed stochastic controller in tracking the realistic commanded attitude profile with a desired standard deviation characteristic. To follow the pre-defined path from launch to main engine cutoff of the X-33, the X-33 high fidelity simulator, MAVERIC [17] issues a set of commanded Euler angles that are scheduled in reference to the magnitude of relative velocity of the vehicle. The desired Euler angle rate is found through the derivative block. The initial condition of the X-33 is set to be 0 3.5o , 0 85o , 0 1o and zero angular velocities [3]. The control parameters are found

to be 0 [0.1,0.1,0.1]T and [3.7369,3.6524,3.7200]T respectively.

Fig. 2(a) through Fig. 2(c) demonstrate the mean value of Euler angle profiles (validated through 10,000 Monte Carlo runs) achieved using the proposed method where the tracking errors are less than 1-2 degrees throughout the launch phase.

0 50 100 150 200 250-25

-20

-15

-10

-5

0

5

Time (s)

Rol

l (de

gre

es)

Command

SC

Fig. 2(a) Mean value of the roll angle history (case 2)

0 50 100 150 200 2500

10

20

30

40

50

60

70

80

90

Time (s)

Pitc

h A

ngl

e (

deg

ree

s)

Command

SC

Fig. 2(b) Mean value of the pitch angle history (case 2)

0 50 100 150 200 2500

2

4

6

8

10

12

14

Time (s)

Yaw

Ang

le (

degr

ees)

Command

SC

Fig. 2(c) Mean value of the yaw angle history (case 2)

The torques applied are shown in Figs. 2(d)-2(f), which is within the saturation level. The chattering phenomena shown in these figures are because of the random noises in the inertia, control torques, and the additive noise to the dynamics model. A carefully designed filter can be used to reduce the chatters, which is not the main purpose of the paper. As already been shown in the simulation case 1, the main advantage of the method is the ability to control the steady state performance distribution. As demonstrated in Figs. 2(g)-2(i), the standard deviations of the attitude profile in simulated X-33 meet the requirement of 0.2o .

American Institute of Aeronautics and Astronautics11

0 50 100 150 200 250-6

-4

-2

0

2

4x 10

4

Time (s)

Rol

l Tor

que

(ft-

lb)

Fig. 2(d) Mean value of the torque in x direction (case 2)

0 50 100 150 200 250-4

-2

0

2

4x 10

4

Time (s)

Pitc

h T

orqu

e (f

t-lb

)

Fig. 2(e) Mean value of the torque in y direction (case 2)

0 50 100 150 200 250-1

-0.5

0

0.5

1

1.5x 10

4

Time (s)

Yaw

Tor

que

(ft-

lb)

Fig. 2(f) Mean value of the torque in z direction (case 2)

0 50 100 150 200 2500

1

2

3

4

Time (s)

Sta

ndar

d D

evia

tion

Rol

l Ang

le (

degr

ee)

Steady State Value: 0.21

Fig. 2(g) Standard deviation of the roll angle (case 2)

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

Time (s)

Sta

ndar

d D

evia

tion

Pitc

h A

ngle

(de

gree

)

Fig. 2(h) Standard deviation of the pitch angle (case 2)

0 50 100 150 200 2500

1

2

3

4

Time (s)

Sta

ndar

d D

evia

tion

Yaw

Ang

le (

degr

ee)

Steady State Value: 0.20

Fig. 2(i) Standard deviation of the yaw angle (case 2)

V. ConclusionIn this paper, a novel quadrature based method of moments has been proposed to propagate the Fokker-Planck

equation accurately and efficiently. Combined with a newly proposed nonlinear robust control (without any discontinuity functions involved), the unknown control parameters in the closed-loop system is found through a constrained nonlinear optimization offline. This method has been used in designing nonlinear stochastic tracking

American Institute of Aeronautics and Astronautics12

control for the ascent phase of the X-33 reusable launch vehicle. The control objectives of tracking the commanded Euler angle profile while achieving desired (attainable) steady state statistical characteristics have been successfully accomplished using the proposed method under functional and parametric uncertainties as well as additive random noise. The advantages of the proposed stochastic control design method which have been validated through 10,000 Monte Carlo runs are: (1) the closed-loop tracking system is asymptotically stable and robust with respect to functional and parametric uncertainties; (2) any (attainable) higher order steady state moments of the state/output variables can be controlled and the system is stable up to the order of the highest moment used in design; (3) the state process can be unknown and is not required to be Gaussian, and (4) no Monte Carlo analysis is required in the design. The results obtained from the proposed method matches very well with the Monte Carlo simulation.

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