An Approach to Evaluate the Region of Attraction of Satellites

controlled by SDRE

ALESSANDRO GERLINGER ROMERO

Space Mechanics and Contro Division

National Institute of Space Research

Astronautas Avenue, 1758 - São José dos Campos

BRAZIL

LUIZ CARLOS GADELHA DE SOUZA

Engineering Center

Federal University of ABC

dos Estados Avenue - São Bernardo do Campo

BRAZIL

Abstract: The control of a satellite can be designed with success by linear control theory if the satellite has slow

angular motions. However, for fast maneuvers, the linearized models are not able to represent the effects of the

nonlinear terms. One candidate technique for the design of the satellite's control under fast maneuvers is the State-

Dependent Riccati Equation (SDRE). SDRE provides an effective algorithm for synthesizing nonlinear feedback

control by allowing nonlinearities. Nonetheless, much criticism has been leveled against the SDRE because it does

not provide assurance of global asymptotic stability. Additionally, there are situations in which global asymptotic

stability cannot be achieved (e.g., systems with multiple equilibrium points). Therefore, especially in aerospace,

estimating the region of attraction (ROA) is fundamental. The Brazilian National Institute for Space Research

(INPE, in Portuguese) was demanded by the Brazilian government to build remote-sensing satellites, such as the

Amazonia-1 mission. In such missions, the satellite must be stabilized in three axes so that the optical payload can

point to the desired target. In this paper, we share an approach to evaluate the ROAs of Amazonia-1 controlled by

LQR (the linear counterpart of SDRE) and SDRE. The initial results showed SDRE has a larger ROA than LQR.

Key-Words: Nonlinear, control, SDRE, LQR, Region of Attraction.

Received: June 22, 2021. Revised: March 17, 2022. Accepted: April 19, 2022. Published: May 7, 2022.

1 Introduction

The design of a satellite attitude and orbit control

subsystem (AOCS) that involves plant uncertainties,

large-angle maneuvers, and fast attitude control fol-

lowing a stringent pointing, requires nonlinear control

techniques in order to satisfy performance and robust-

ness requirements. An example is a typical mission

of the Brazilian National Institute for Space Research

(INPE), in which the AOCS must stabilize a satellite

in three axes so that the optical payload can point to

the desired target with few arcsecs of pointing accu-

racy.

One candidate technique for a nonlinear AOCS

control law is the State-Dependent Riccati Equation

(SDRE). SDRE is based on the arrangement of the

system model in a form known as state-dependent co-

efficient (SDC) matrices. Accordingly, a suboptimal

control law is carried out by a real-time solution of

an algebraic Riccati equation (ARE) using the SDC

matrices by means of a numerical algorithm.

SDRE was originally proposed by [1] and then ex-

plored in detail by [2]. A good survey of the SDRE

technique can be found in [3] and its systematic appli-

cation to deal with a nonlinear plant in [4]. The SDRE

technique was applied by [5, 6, 7, 8, 9] for controlling

a nonlinear system similar to the six-degree of free-

dom satellite model considered in this paper.

The application of the SDRE technique, and, con-

sequently, the ARE problem that arises, have already

been studied in the available literature, e.g., [10] in-

vestigated the approaches for the ARE solving as well

as the resource requirements for such online solving.

Recently, [6] proposed the usage of differential alge-

bra to reduce the resource requirements for the real-

time implementation of SDRE controllers. In fact, the

intensive resource requirements for the online ARE

solving is the major drawback of SDRE. Nonetheless,

the SDRE has three major advantages: (a) simplic-

ity, (b) numerical tractability, and (c) flexibility for

the designer, being comparable to the flexibility in the

LQR [6].

The origin of an SDRE controlled system is a lo-

cally asymptotically stable equilibrium point [2]. Fur-

thermore, the knowledge of its region of attraction is

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fundamental due to the local stability even more in the

presence of nonlinearities.

Indeed, much criticism has been leveled against

the SDRE technique since it does not provide assur-

ance of global asymptotic stability. However, empir-

ical experience shows that in many cases the region

of attraction (ROA) may be as large as the domain of

interest [3]. Moreover, there are situations in which

global asymptotic stability cannot be achieved (for

example, systems with multiple equilibrium points).

Therefore, especially in aerospace, estimating the re-

gion of attraction is fundamental [3].

Obtaining a good estimate of such a ROA, espe-

cially of a higher-order system, is a challenging task

in itself. In fact, [11] states that analytical ROA for

nonlinear systems with dimensions greater than two is

usually unavailable. Such a task becomes even more

difficult since the closed-loop matrix of SDRE is usu-

ally not available in the closed form [12].

The well-known Lyapunov approaches to estimate

the region of attraction cannot be used for SDRE

since the closed-loop system equations are usually

not known explicitly. [12, 13] proposed procedures

to reduce the effort of ROA’s computation focusing

on the maximum and the minimum values of feed-

back gains over a chosen region of the statespace. The

other alternative is to make time-domain simulations

of the closed-loop system, which is cumbersome and

costly [12, 13].

In this paper, we share an approach to evaluate

the ROA of Amazonia-1 controlled by LQR (linear-

quadratic regulator, the linear counterpart of SDRE)

and SDRE. The initial results showed SDRE has a

larger ROA than LQR in the presence of nonlineari-

ties. Recall LQR guarantees global stability for linear

systems (afterward, the linearization process), never-

theless, such a property is lost when nonlinearities are

accounted for.

In Section 2, the problem description is presented.

In Section 3, the satellite physical modeling is re-

viewed. In Section 4, we explore the state-space

model, the controllers, and the approach for the

ROAs. In Section 5, we share simulation results. Fi-

nally, the conclusions are shared in Section 6.

2 Problem Description

The SDRE technique entails factorization (that is,

parametrization) of the nonlinear dynamics into the

state vector and the product of a matrix-valued func-

tion that depends on the state itself. In doing so,

SDRE brings the nonlinear system to a (nonunique)

linear structure having SDC matrices given by Eq. (1).



˙x=A(x)x +B(x)u

y =Cx (1)

where x ∈Rnis the state vector and u ∈Rmis the

control vector. Notice that the SDC form has the same

structure as a linear system, but with the system ma-

trices, Aand B, being functions of the state vector.

The nonuniqueness of the SDC matrices creates ex-

tra degrees of freedom, which can be used to enhance

controller performance, however, it poses challenges

since not all SDC matrices fulfill the SDRE require-

ments, e.g., the pair (A,B) must be pointwise stabiliz-

able.

The system model in Eq. (1) is subject of the cost

functional described in Eq. (2).

J(x0, u) = 1

2∞

(xTQ(x)x +uTR(x)u)dt (2)

where Q(x)∈Rn×nand R(x)∈Rm×mare the

state-dependent weighting matrices. In order to en-

sure local stability, Q(x)is required to be positive

semi-definite for all x and R(x)is required to be pos-

itive for all x [10].

The SDRE controller linearizes the plant about the

current operating point and creates constant states-

pace matrices so that the LQR can be used. This pro-

cess is repeated in all samplings steps, resulting in a

pointwise linear model from a non-linear model, so

that an ARE is solved and a control law is computed

also in each step. Therefore, according to LQR the-

ory and Eq. (1) and (2), the state-feedback control law

in each sampling step is u =−K(x)x and the state-

dependent gain K(x)is obtained by Eq. (3) [4].

K(x) = R−1(x)BT(x)P(x)(3)

where P(x)is the unique, symmetric, positive-

definite solution of the algebraic state-dependent Ric-

cati equation (SDRE) given by Eq. (4) [4].

P(x)A(x) + AT(x)P(x)

−P(x)B(x)R−1(x)BT(x)P(x)

+Q(x) = 0

(4)

Considering that Eq. (4) is solved in each sampling

step, it is reduced to an ARE. Finally, the conditions

for the application of the SDRE technique in a given

system model are [4]:

1. A(x)∈C1(Rw)

2. B(x), C(x), Q(x), R(x)∈C0(Rw)

3. Q(x)is positive semi-definite and R(x)is posi-

tive definite

4. A(x)x=⇒A(0)0 = 0, i.e., the origin is an

equilibrium point

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5. pair(A, B)is pointwise stabilizable (a sufficient

test for stabilizability is to check the rank of con-

trollability matrix)

6. pair(A, Q1

2)is pointwise detectable (a sufficient

test for detectability is to check the rank of ob-

servability matrix)

2.1 Region of Attraction

One is frequently interested in the region of attraction

(ROA; also called domain of attraction or basin of at-

traction), i.e., the region of the statespace in which

the initial conditions of the trajectories lie in order to

attain stable behavior [14].

Equilibrium points, if exist, must lie in the regions

of attraction, indeed, a state xe∈ Rnis an equilib-

rium point of a nonlinear dynamical system ˙x=f(x)

if x0=xe=⇒x(t) = xe,∀t > 0so f(xe) = 0.

A nonlinear system can have an infinite number of

equilibrium points, and, each one can be stable or

unstable. Additionally, to investigate the stability of

a particular equilibrium point x0it is convenient to

transform the equilibrium point to the origin x∗= 0

through the transformation x∗=x−x0[14].

Such equilibrium points, if exist, lie in the regions

of attraction, which are defined by their attractors.

An attractor is a subset A∈ Rnof the statespace

characterized by the following three conditions: (i)

a∈A=⇒f(a, t)∈A, ∀t > 0; (ii) there exists

a vicinity of A(region of attraction) which consists

of all trajectories that enter Afor t→ ∞; and, (iii)

there is no proper (non-empty) subset of Ahaving

the first two properties. The attractors can be clas-

sified in: (I) Fixed point - the final state that a dy-

namic system evolves towards corresponds to an at-

tracting fixed point, i.e., stable equilibrium point; (II)

Limit cycle - a periodic orbit; (III) Quasiperiodic - it

exhibits irregular periodicity; (IV) Strange attractor -

when such sets cannot be easily described.

Regarding the fixed point attractor, it can be de-

fined by the following equation [15].

A={x0∈ Rn:lim

t→+∞

x(t, x0) = 0}(5)

It is important for practical reasons to have infor-

mation about the size and/or the shape of A. Indeed,

the stability properties could be of scarce utility if the

region of attraction is very small, or if the equilib-

rium point is very close to its boundary. There is a

wide literature about theoretical methods for the de-

termination of A, and about numerical methods for its

approximate estimation [12, 15, 13].

3 Satellite Physical Modeling

Focusing on a typical mission developed by INPE, the

AOCS must stabilize a satellite in three axes so that

the optical payload can point to the desired target. The

next subsections explore the kinematics and the rota-

tional dynamics of the satellite attitude available in

the simulator.

3.1 Kinematics

Given the ECI reference frame (Fi) and the frame

defined in the satellite with origin in its centre of

mass (the body-fixed frame, Fb), then a rotation R∈

SO(3) (SO(3) is the set of all attitudes of a rigid

body described by 3×3orthogonal matrices whose

determinant is one) represented by a unit quaternion

Q= [q1q2q3|q4]Tcan define the attitude of the

satellite.

Defining the angular velocity ω = [ω1ω2ω3]Tof

Fbwith respect to Fimeasured in the Fb, the kinemat-

ics can be described by Eq. (6) [16].

Q=1

2Ω(ω)Q

Ω(ω)≜





0ω3−ω2ω1

−ω30ω1ω2

ω2−ω10ω3

−ω1−ω2−ω30







(6)

where the unit quaternion Qsatisfies the following

identity:

1+q2

2+q2

3+q2

4= 1 (7)

Eq. (6) allows the prediction of the satellite’s atti-

tude if it is available the initial attitude and the history

of the change in the angular velocity ( ˙

Q=F(ω, t)).

Another possible derivation of the Eq. (6) is using

the vector g (Gibbs vector or Rodrigues parameter)

as Q= [gT|q4].

Q=−1

2ω×

ωTq1

q3+1

2q413×3

0ω (8)

where ω×is the cross-product skew-symetric matrix

of ω and 1is the identity matrix. Note the Gibbs vec-

tor is geometrically singular since it is not defined

for 180◦of rotation [17], nonetheless, the Eq. (8) is

global.

3.2 Rotational Dynamics

The satellite has a set of 3 reaction wheels, each one

aligned with its principal axes of inertia, moreover,

such type of actuator, momentum exchange actuators,

does not change the angular momentum of the satel-

lite. Consequently, it is mandatory to model their in-

fluence in the satellite, in particular, the angular mo-

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mentum of the satellite is defined by Eq. (9).



h= (I

I−



n=1

In,sanaT

n)ω +



n=1

hw,n an(9)

where I

Iis the inertia tensor, In,s is the inertia mo-

ment of the nreaction wheel in its symmetry axis an,

hw,n is the angular momentum of the nreaction wheel

about its centre of mass (hw,n =In,saT

nω+In,sωn)

and ωnis the angular velocity of the nreaction wheel.

One can define Ib=I

I−3

n=1 In,sanaT

n. Using

Ib, the motion of the satellite is described by Eq. (10).

Ib˙ωb=gcm −ω×(Ibω+



n=1

hw,nan)−



n=1

gnan

(10)

where gcm is the net external torque and gnare the

torques generated by the reactions wheels ( ˙

hw,n =

gn).

4 Controller Design and ROA

Two dynamics states must be controlled: (1) the atti-

tude (modeled as unit quaternions Qby Equation (6)

or Equation (8)) and (2) its stability ( ˙

Q, in other

words, the angular velocity ωof the satellite by Equa-

tion (10)). The following subsections explore the

state-space modeling and the synthesis of controllers.

4.1 Linear Control based on LQR

Equation (6) and Equation (7) can be used to linearize

the system around a stationary point (ω= 0 and Q=

[0 0 0 −1]T, assuming there is no net external torque,

gcm = 0), which leads to Equation (11).

x1

x2=Q

ω

˙x1

˙x2=



0−1

2I3x3

0 0

0 0 

x1

x2+0

−I−1

b[u1]

[y] = Ix1

x2

(11)

Equation (11) defines the constant matrixes A,

B and C for a linear state-space representation of

Amazonia-1 (see Table 1) around the stationary point.

However, the constant matrices A and B are not

stabilizable since the controllability matrix of the

pair(A,B) has no full rank. Indeed, [18] showed that

this linearized model with all quaternion components

was not stabilizable, meaning that LQR is not appli-

cable.

Equation (7) defines a direct method to find q4,

therefore, one option is to model the state of the sys-

tem without such component of the quaternion [18],

which leads to the following equation:

x3

x2=











˙x3

˙x2=0−1

2I3x3

0 0 x3

x2+0

−I−1

b[u1]

[y] = Ix3

x2

(12)

Equation (12) defines the constant matrixes A, B

and C for an alternative linear state-space representa-

tion of Amazonia-1 (see Table 1) around the station-

ary point. In such the statespace, constant matrices A

and B are stabilizable.

4.2 Nonlinear Control based on SDRE

Assuming that there are no net external torques

(gcm = 0), the statespace model can be defined us-

ing Eq. (6) (Ω) and (10), however, the SDC matrices

do not fulfill the SDRE requirements, in particular,

the pair (A,B) is not pointwise stabilizable.

An option for the definition of the SDC matrices

is to use Eq. (8), which leads to Eq. (13).

˙x1

˙x2=





−1

2ω×

ωT01

2q4I3×3

0

0 0 −I−1

bω×Ib+I−1

b(3

n=1 hw,nan)×



x0

x2+0

−I−1

b[u1]

[y] = Ix0

x2

(13)

Eq. (13) has been shown to satisfy SDRE condi-

tions described in Section 2.

Equation (13) has been shown to satisfy SDRE

conditions in the majority of statespace with excep-

tion of the region on which the angular velocity is

close to 0(the pair(A,B) loses rank in such region).

In practical problems, regarding such a region, one

approach is to switch to another SDC parametriza-

tion [11] or to resort to LQR. Note such the known

limitation of this particular parametrization of SDRE

imposes laxity.

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Table 1: Satellite characteristics, initial conditions and references.

Name Value

Satellite Characteristics

inertia tensor (kg.m2), I

I310.0 1.11 1.01

1.11 360.0−0.35

1.01 −0.35 530.7

Actuators Characteristics - Reaction Wheels

inertia tensor of 3 reaction wheels (kg.m2), 3

n=1 In,sanaT

ndiag(0.01911,0.01911,0.01911)

maximum torque (N.m) 0.075

maximum angular velocity (RP M ) 6000

Initial conditions

attitude (degrees, XYZ) [0 0 180]T

angular velocity (radians/second, XYZ) [000.024]T

References for the controller

solar vector in the body (XYZ) [100]T

angular velocity (radians/second, XYZ) [000]T

4.3 Region of Attraction

Firstly, the control laws, independent of the applied

technique being linear or nonlinear, are modeled

and too many time-domains simulations are avail-

able for their analysis. Furthermore, in the presence

of nonlinearities even the techniques that guarantee

global asymptotically stability can lose such property.

Therefore, the main task is to assess the presence of a

fixed point attractor as well as its size and shape.

Nonetheless, it is somewhat difficult to depict the

ROA for statespace systems with dimensions higher

than three. Facing such difficulty, [11] chose to list

the domain of interest and to plot a small subset of

simulations. Such an approach offers restricted sup-

port for the comparison of different ROAs.

Inspired by the works of Henri Poincaré, in par-

ticular, Poincaré maps [19] which defines a lower-

dimensional subspace for qualitative analysis. The

present paper applied two euclidean norms, namely

L2-norm of Euler angles and L2-norm of angular ve-

locities, to define a two-dimensional space for quali-

tative and quantitative analysis. In such a way that the

area of the ROA (dimensionless quantity) can be ana-

lytically computed and compared; and, the plot of the

ROA can allow straightforward qualitative analysis.

Besides, the definition of the fixed point attractor

presented in Equation (5) is restricted by the defini-

tion of an explicit final time tfand a numerical error

ϵ, according to the following equation:

A={x0∈ Rn:lim

t→tf

||x(t, x0)||2< ϵ}(14)

Equipped with (A) the two-dimensional space

for qualitative analysis of the original up to seven-

dimensional statespace, and (B) Equation (14), simple

polygons (they do not intersect themselves and have

no holes) of ROAs are defined in the two-dimensional

space for the initial conditions x0. The area of such

polygons is the main measure for the quantitative

comparison of different control laws.

Taking into account the limitations of the control

laws previously stated in Subsection 4.2, the defini-

tion of the fixed point attractor presented in Equa-

tion (14) is once restricted by the selection of sole an-

gular velocities (x2in Equation (11), Equation (12),

and Equation (13)), ω-stability in [16], according to

the following equation:

A={x0∈ Rn:lim

t→tf

||x2(t, x0)||2< ϵ}(15)

Focusing on the area of ROAs A, one defines a

Monte Carlo perturbation model for a given initial

conditions, performs the time-domain simulation un-

til the predefined tf, and, finally, computes the mea-

sure.

5 Simulation Results

The results were computed using the satellite (mod-

eled as a nonlinear system) and the control laws

(LQR, and SDRE) with nonlinearities in the reaction

wheels, i.e., maximum torque and maximum angu-

lar velocity. Furthermore, such the results were ob-

tained running a full Monte Carlo perturbation model

described as follows.

Regarding initial attitudes defined by 3-2-1 Eu-

ler angles (Z-Y-X, nonclassical Euler angles), it is

well-known that representing Ybeyond ±90 degrees

(that means ±180 degrees) would give two Euler an-

gles solution for every rotation, so Yis limited to

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±90 degrees. Xand Zare limited to ±180 degrees.

Therefore, independent distributions are applied for

each Euler angle respecting the limits previously dis-

cussed in order to define the initial attitudes (3-2-1, Z-

Y-X, nonclassical Euler angles, which are converted

in quaternions) in a given Monte Carlo perturbation

model.

Regarding initial angular velocities, they are de-

fined by independent distributions based on the max-

imum angular velocity of the satellite that is con-

trollable by the reaction wheels. Using Section 3.2

and Table 1, the maximum angular momentum of

the set of reactions wheels was computed ( 

hwmax =



Iw. ωwmax ) and then the corresponding maximum an-

gular velocity of the satellite was found solving a ma-

trix equation ( 

hwmax =

I. ωmax). The result in radians

per second was ωmax ={0.0385,0.0332,0.0225},

the norm (L2) was 0.0556, and the infinity norm was

0.0385, which is the value used as parameter in Ta-

ble 2.

In summary, the approach for the evaluation of the

region of attraction can be summarized as:

• Compute initial conditions for the Monte Carlo

perturbation model

Using independent distributions, compute

the 3-2-1 Euler angles (Z-Y-X, nonclassical Eu-

ler angles) in the range (±180,±90,±180)

Using independent distributions, compute

the angular velocities based on the maximum an-

gular velocity of satellite that is controllable by

the reaction wheels

• Perform the time-domain simulation until the

predefined tf

• Compute ROAs

The results are based on the satellite Amazonia-

1, which is characterized by Table 1, furthermore, the

simulations were conducted with the full Monte Carlo

perturbation model tuned with the parameters shared

in Table 2.

The initial conditions uniformly distributed com-

puted using such parameters by the Monte Carlo per-

turbation model are depicted using a two-dimensional

space, in which the norm of Euler angles is along the

X-axis and the norm of angular velocities is along the

Y-axis for each initial condition. This space has its

bounds constrained by the norm of Euler Angles rang-

ing from 0 to 270 degrees and the norm of angular ve-

locities ranging from 0 to 0.066 radians per second,

following the parameters presented in Table 2. How-

ever, as there is no mechanism to desaturate the re-

actions wheels, the limit expected for the norm of the

angular velocities of the initial conditions inside any

ROA is the previously shared 0.0556 radians per sec-

ond (in the presence of the nonlinearities). The same

two-dimensional space is used to depict ROAs.

Figure 1 shows the initial conditions uniformly

distributed by such a Monte Carlo perturbation model

execution (400 simulations, two different control laws

for each initial condition).

Taking into account the applied nonlinearities in

the reaction wheels (maximum torque, and maximum

angular velocity), for any step of a given maneuver

the actual control torque generated by the reaction

wheels could be smaller than the computed control.

Indeed, with such nonlinearities, the actual control

was smaller than the computed control in the simu-

lations as shown in Figure 2. Note points are close to

the X-axis in the sense that actual control was much

smaller than the bisectrix (where actual equals com-

puted control).

The result of such a “lack of control” is depicted

in Figure 3 that shows the ROA of LQR (in blue, leg-

end ProportionalLinearQuatuernionPartialLQRCon-

troller), and SDRE (in red, legend ProportionalNon-

LinearQuaternionSDREController_GIBBS).

The number of samples that converged by Equa-

tion (15) is less than the initial conditions, therefore,

the global asymptotic stability property is not present

in LQR or SDRE. Nonetheless, the ROA of the SDRE

is larger since more initial conditions converged for

tf. Moreover, both control laws are able to control

the whole range of Euler angles, consequently, the

angular velocities are the critical aspects to be con-

strained in the initial conditions. Such the fact corrob-

orates ω-stability [16] as the major concern in accor-

dance with Equation (15). Furthermore, it confirms

the common sense the controlling of angular veloc-

ities is constrained by nonlinearities in the actuators

leading to saturation.

6 Conclusion

As the results are based on analysis through simula-

tions, they are neither valid for general cases nor for

scenarios out of the range of the Monte Carlo pertur-

bation models due to the underlining nonlinear dy-

namics.

Although the results provide empirical evidence

that SDRE has a larger ROA than LQR in the case of

Amazonia-1, the major contribution of the paper is the

approach for the evaluation of the ROAs since such

the approach provides a tractable numerical algorithm

to compare control techniques. Moreover, confidence

turns to be a matter of computational power.

References:

[1] J. D. Pearson, “Approximation methods in op-

timal control i. sub-optimal control,” Journal of

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Table 2: Monte Carlo perturbation model parameters.

Name Value

3-2-1 Euler angles (degrees) Z:U(−180,180)

Y:U(−90,90)

X:U(−180,180)

angular velocities (rad/s) X:U(−0.0385,0.0385)

Y:U(−0.0385,0.0385)

Z:U(−0.0385,0.0385)

Q I

R I

ϵ(rad/s) 0.0001

samples 200

tf(seconds) 3600

fixed step size (seconds) 0.05

Electronics and Control, vol. 13, no. 5, pp. 453–

469, 1962.

[2] J. R. Cloutier, C. N. D’Souza, and C. P. Mracek,

“Nonlinear regulation and nonlinear H-infinity

control via the state-dependent Riccati equation

technique.,” Conference on Nonlinear Problems

in Aviation, 1996.

[3] T. Çimen, “State-Dependent Riccati Equation

(SDRE) control: a survey,” IFAC Proceedings

Volumes (IFAC-PapersOnline), vol. 17, no. 1 pt.

1, pp. 3761–3775, 2008.

[4] T. Çimen, “Systematic and effective design of

nonlinear feedback controllers via the state-

dependent Riccati equation (SDRE) method,”

Annual Reviews in Control, vol. 34, no. 1,

pp. 32–51, 2010.

[5] R. G. Gonzales and L. C. G. d. Souza, “Ap-

plication of the sdre method to design a atti-

tude control system simulator,” Advances in the

Astronautical Sciences, vol. 134, no. Part 1-3,

pp. 2251–2258, 2009.

[6] G. DiMauro, M. Schlotterer, S. Theil, and

M. Lavagna, “Nonlinear control for proxim-

ity operations based on differential algebra,”

Journal of Guidance, Control, and Dynamics,

vol. 38, pp. 2173–2187, apr 2015.

[7] A. G. Romero, L. C. G. Souza, and R. A. Cha-

gas, Application of the SDRE Technique in the

Satellite Attitude and Orbit Control System with

Nonlinear Dynamics. 2018.

[8] A. G. Romero and L. C. G. Souza, “Application

of the sdre technique based on java in a cube-

sat attitude and orbit control subsystem,” in Pro-

ceedings..., IAA Latin American CubeSat, De-

cember 2018.

[9] A. G. Romero, Application of the SDRE tech-

nique in the satellite attitude and orbit control

system with nonlinear dynamics. PhD thesis, In-

stituto Nacional de Pesquisas Espaciais (INPE),

São José dos Campos, 2021-12-07 2022.

[10] P. K. Menon, T. Lam, L. S. Crawford, and V. H.

Cheng, “Real-time computational methods for

SDRE nonlinear control of missiles,” Proceed-

ings of the American Control Conference, vol. 1,

pp. 232–237, 2002.

[11] J. Yao, Q. Hu, and J. Zheng, “Nonlinear opti-

mal attitude control of spacecraft using novel

state-dependent coefficient parameterizations,”

Aerospace Science and Technology, vol. 112,

p. 106586, 2021.

[12] E. Erdem and A. Alleyne, “Estimation of stabil-

ity regions of sdre controlled systems using vec-

tor norms,” in Proceedings..., vol. 1, pp. 80–85

vol.1, American Control Conference, 2002.

[13] A. Bracci, M. Innocenti, and L. Pollini, “Es-

timation of the region of attraction for state-

dependent riccati equation controllers,” Journal

of Guidance, Control, and Dynamics, vol. 29,

no. 6, pp. 1427–1430, 2006.

[14] P. C. Parks and V. Hahn, Stability Theory. New

York: Prentice-Hall, 1992.

[15] L. Bacciotti, A.; Rosier, Liapunov functions and

stability in control theory. Berlin, Germany:

Springer, 2 ed., 2005.

[16] P. C. Hughes, Spacecraft attitude dynamics.

New York, 1986.

[17] P. W. Fortescue and G. G. Swinerd, “Atti-

tude control,” Spacecraft Systems Engineering,

pp. 289–326, 2011.

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[18] Y. Yang, “Analytic LQR design for spacecraft

control system based on quaternion model,”

Journal of Aerospace Engineering, vol. 25,

no. 3, pp. 448–453, 2012.

[19] H. Leipholz, Stability theory: an introduction to

the stability of dynamic systems and rigid bod-

ies. Academic Press, 1970.

Contribution of individual authors to

the creation of a scientific article

(ghostwriting policy)

Alessandro Gerlinger Romero carried out the model-

ing, the code developing, the simulation and the writ-

ing.

Luiz Carlos Gadelha de Souza oriented the first author

and contributed in the writing.

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License 4.0 (Attribution 4.0

International , CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.en_US

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Figure 1: Initial conditions.

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Figure 2: Actual versus computed control with nonlinearities in the reaction wheels.

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Figure 3: ROA with nonlinearities in the reactions wheels.

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