Suppression of Chaotic Modes in Spacecraft

with Asymmetric Actuator Constraints

Carlos M. N. Velosa, and Kouamana Bousson

LAETA-UBI/AeroG Research Group

Avionics and Control Laboratory

Department of Aerospace Sciences

University of Beira Interior

6201-001 Covilhã, Portugal

ABSTRACT The present paper proposes a robust and easily implementable control technique for the output control of

nonlinear systems which take into account asymmetric magnitude and rate actuator constraints. The

technique consists in decomposing the nonlinear system into a linear part plus a nonlinear part and in

designing, with the help of an auxiliary system, a control law to stabilize the resultant augmented system

through and only through its linear part. The technique is formulated in such a way that it is able to

control nonlinear systems that may exhibit very complex behaviours, and therefore, one considers that it

is an asset to supress undesirable chaotic motions which may arise in spacecraft. Numerical simulations

are performed to validate the effectiveness of the technique proposed. It is applied in two real aerospace

systems: first, to control the attitude of a spacecraft under the effect of external disturbances, and

secondly, to control the position of a spacecraft in the restricted three-body problem Earth-Moon-

Spacecraft. The results approve the success of the technique, and this work contributes effectively for the

most advanced techniques released to community, in the sense that it provides a control strategy which

assumes asymmetric magnitude and rate constraints, as opposed to other control techniques which do not

consider any kind of constraints and that are particularly required in the aerospace systems.

Keywords: Chaos, Spacecraft, Output regulation, Constrained control, Asymmetric constraints.

1. INTRODUCTION

Very recently there has been a large effort, either by the industry either by the academies, in developing

control techniques for dynamical systems that may exhibit extremely complex behaviours. The effort

arises because scientists and engineers have realized over the past years that a given system may come to

exhibit undesirable and complex dynamics even if the system has been designed, through an appropriate

choice of its parameters, to exhibit a regular behaviour. Indeed, that is what happens with the well-known

chaotic systems. Chaotic systems are deterministic nonlinear systems, most of them even with a relatively

simple structure, but that, despite being governed by well-defined dynamical laws, exhibit unpredictable

and very complex motions. Its main characteristics are the high sensitivity to initial conditions and the

high sensitivity to parameter changes, and that is why the behaviour exhibited may be completely

different than expected: a tiny variation in the parameters is enough to trigger a quite different behaviour.

On the other hand and in the same line of thought, it can be proven, either through analytical

techniques, numerical techniques or experimental tests, that any nonlinear system may exhibit chaotic

oscillations if the system is exposed to external disturbances with particular characteristics. A small

disturbance with specific amplitude and frequency(ies) is enough to trigger a chaotic dynamics. By way

of example, the attitude of a rigid satellite in a circular orbit near the equatorial plane of the Earth, despite

being designed to exhibit a regular motion, may become chaotic if the satellite contains magnetic

elements. The terrestrial magnetic field acts as perturbing torques that are sufficient to trigger a chaotic

attitude (Chen & Liu, 2002; Cheng & Liu, 1999). It has been reported also that in some circumstances

chaos may appear in dynamical systems if bounded controls are taken into account. If the controlled

system is subject to actuator constraints the limited input signals can be seen as perturbing terms

satisfying “particular characteristics” and cause structural instabilities (Demenkov, 2008).

Chaos may appear undesirably in several applications of the aerospace- and aeronautical- domain. It

has been shown, mainly by the Melnikov function and by the Lyapunov exponents, that chaos appears in

some spacecraft models such as in spinning satellites, gyrostat satellites, tethered satellites, and other

complicated satellites (Liu & Chen, 2003; Liu & Liqun, 2013). In aeronautics, the presence of chaos in

aircraft wings, horizontal- and vertical- stabilizers have been also reported either in subsonic flow as in

supersonic flow, and it is a topic that has been extensively studied due to the catastrophic consequences

that may be caused by chaotic vibrations (Alstrom, Marzocca, Bollt, & Ahmadi, 2010; Bousson &

Velosa, 2014a; Bousson, 2010; Wang, Chen, & Yau, 2013). Chaotic vibrations and limit cycle

oscillations (flutter) are dangerous phenomena that can lead to structural failure due to material fatigue

and posteriorly to an aircraft disaster in the worst scenario.

Techniques designed specifically for the control of chaotic systems such as the renowned OGY

technique (Ott, Grebogi, & Yorke, 1990), the Continuous Delayed Feedback Control (Pyragas, 1992) and

the Otani-Jones Control (Otani & Jones, 1997) proved to be successful to supress undesired chaotic

oscillations. However, none of these techniques takes into account actuator constraints, and it is well-

known that any control system must to consider them. The thrust of a rocket is bounded, the ranges of the

control surfaces of an aircraft are bounded, the voltages applied to electronic circuits are bounded, and so

on. On the other hand, some demanding applications require also actuator rate constraints. Valves and

pumps have maximum throughputs, the responsiveness of pneumatics is bounded, operational amplifiers

have maximum slew-rates, and so on, and if these constraints are not considered when formulating the

control law the control signals can lead to catastrophic scenarios. Interesting examples about aircraft and

nuclear disasters caused by actuator limitations can be found in (Murray, 1999; Stein, 2003).

A common approach to ensure that the control is bounded consists in adding a saturation block after

the controller output and resort to anti-windup techniques to prevent both the controller- and/or plant-

windup phenomenon (Hippe, 2006). However, the approach is not the best because it introduces

additional nonlinearities in the system, degrades its performance, and can even lead it to the instability.

Several advanced control techniques and some of them even quite sophisticated have been used to

control nonlinear systems including those that may exhibit chaotic behaviours. Feedback Linearization,

Backstepping Control, Robust Control (SMC - Sliding Mode Control, LQG/LTR - Linear-Quadratic-

Gaussian/Loop Transfer Recovery, QFT - Quantitative Feedback Theory, 𝜇-Synthesis Control, 𝐻2/𝐻∞),

Adaptive Control (MRAC - Model Reference Adaptive Control, STR - Self-Tuning Regulator, Gain

Scheduling, Dual Control), Predictive Control (MPC/RHC - Model Predictive Control/Receding Horizon

Control), and control based on Linear Matrix Inequalities (LMI’s), are the most cited techniques available

in literature (Bousson & Velosa, 2014a). However, all techniques proposed present, that is authors’

knowledge to date, a big drawback in common: constraints simultaneously on the control variables and on

its rates are not taken into account.

Given the characteristics of the chaotic systems, the techniques of control for dynamical systems liable

to exhibit chaotic oscillations must to be robust against external disturbances and to parameter

uncertainties in order to attain an effective control. On the other hand, once it comes to the control of real-

world applications any technique must to consider also actuator constraints. At least, take into account

magnitude and rate constraints which are the constraints that most applications require.

The main purpose of the paper concerned is to propose an easily implementable and robust control

technique for nonlinear dynamical systems that may exhibit undesirable behaviours including the very

complex ones as chaos, and, most important of all, that must to take into account asymmetric magnitude

and rate control constraints. The work is motivated by the fact that any well designed control system must

to deal inevitably with actuator constraints, especially the control systems applied in advanced control

applications as in the aerospace and aeronautical domain, and on the other hand by the fact that it must to

be able to supress even unexpected chaotic motions. In this context, one believes that the present work

contributes with an elegant solution.

The structure of the paper is as follows: section 2 states the problem to be solve in a concise way with

the mathematical formalities; section 3 presents the proposed solution to the problem, which is an

extension of the work previously proposed by (Bousson & Velosa, 2014b). Starting from a technique to

control a nonlinear system through its linear part with the aid of an exosystem, it is carried out a

generalization of the work presented by (Hippe, 2006), that deals with symmetric magnitude and rate

constraints in the case of SIMO (Single-Input, Multiple-Outputs) systems, for the case of MIMO

(Multiple-Inputs, Multiple-Outputs) systems, and a modification is made a posteriori to achieve

asymmetric constraints. In section 4, numerical simulations are performed to prove the effectiveness of

the approach. Two applications are considered. First, the technique is used to control the attitude of a

spacecraft under certain disturbances, and secondly, to control the position of a spacecraft in a restricted

three-body problem. In the latter, it is assumed that the spacecraft describes a chaotic trajectory in space

and the control is posteriorly applied to steer the spacecraft to one of the Lagrangian points of the system

Earth-Moon-Spacecraft. In section 5, future research directions are addressed, and the paper ends, lastly,

in section 6 with a discussion and conclusion about the technique proposed.

2. PROBLEM STATEMENT

Consider the problem of output control in a continuous-time domain of a time-invariant nonlinear system

described by dynamical equations of the form (1). Let the reference outputs, intended to be followed by

the outputs of this controlled system, be generated through a nonlinear system described similarly by

differential equations of the form (2):

Controlled system: �̇� = 𝑓(𝑥, 𝜃) + 𝐵𝑢𝑦 = 𝐶𝑥

(1)

Reference system: �̇�𝑟 = 𝑓𝑟(𝑥𝑟, 𝜃𝑟)𝑦𝑟 = 𝐶𝑟𝑥𝑟

(2)

where 𝑥 ∈ ℝ𝑛, 𝑥𝑟 ∈ ℝ𝑛𝑟 represent the state vectors, 𝜃 ∈ ℝ𝑝, 𝜃𝑟 ∈ ℝ𝑝𝑟 vectors of parameters, 𝑦, 𝑦𝑟 ∈ ℝ𝑞

the outputs of each system (vectors with same dimension), 𝑢 ∈ ℝ𝑚 the control input vector, 𝐶 ∈ ℝ𝑞×𝑛,

𝐶𝑟 ∈ ℝ𝑞×𝑛𝑟 the output matrices, 𝐵 ∈ ℝ𝑛×𝑚 the input matrix, 𝑓, 𝑓𝑟 two nonlinear functions, and �̇� = 𝑑/𝑑𝑡.

Several are the control techniques available to deal with the problem of output control, relying most of

them on the error dynamics. However, when it comes to real control applications wherein actuator

constraints must be taken into account, few are the techniques that one can find in literature to handle

such problem. It is well-known that any physical system of the real world is subject to actuator constraints

and, therefore, the design of an appropriate control technique must consider from the scratch control

saturations, otherwise the actuator limitations can lead to a degradation of the system performance or even

to a disaster in the worst scenario.

A large majority of applications require constraints only on the magnitudes of the control. For those cases,

a typical strategy consists in adding a saturation block at the controller output and resort to anti-windup

techniques to prevent controller- and/or plant- windups. Nevertheless, despite of being a workable

solution, the approach is not the best because it introduces extra nonlinearities in the system and the

resulting performance is not the best. Another vast majority of applications require not only magnitude

control constraints but also rate constraints. By way of example, the operational amplitude range of the

control surfaces of an aircraft is bounded, the thrust of a rocket is bounded, the speed of an electrical

motor is bounded, the voltages that can be applied to an electrical circuit are bounded, and so on.

Some more specific applications, and a special emphasis for the aerospace and aeronautical systems,

require not only control magnitude constraints but also control rate constraints. It is well-known that the

responsiveness of a pneumatic is bounded, valves and pumps have maximum throughputs, the slew-rate

of an operational amplifier is bounded, and so on. When it comes to the control taking into account both

magnitude and rate actuator constraints, and, moreover, to be applied in nonlinear systems (apart from

nonlinearities due to actuator saturations), very few publications are available in literature to solve the

problem.

(Bousson & Velosa, 2014a, 2014b) present two distinct approaches to cope with the control of

nonlinear systems subject both to magnitude and rate actuator constraints, particularly for the class of

chaotic systems. However, the constraints considered in both works are of the form ‖𝑢(𝑡)‖∞ ≤ 𝜂 and

‖�̇�(𝑡)‖∞ ≤ 𝜈 , with 𝜂 ≥ 0 and 𝜈 ≥ 0 denoting the saturation limits, that is, the constraints are

symmetrical with respect to the origin. Although either one of the approaches cover a wide range of

applications, there are some demanding applications that require yet a constrained control but

asymmetrical relative to the origin. Notice for example that the deflection of a control surface of an

aircraft may be between [−25, 35]° (asymmetrical) instead of [−30, 30]° (symmetrical) as in the most

aircraft. The control torques in a spacecraft around a specified axis may be for example 200 Nm in the

clockwise direction and just 100 Nm in the counterclockwise direction. These are asymmetrical

constraints that can be imposed due to mechanical or electrical limitations, or that can arise unexpectedly

for instance due to ice accumulation on the aircraft wing or in case of failure in one of the attitude control

thrusters of the spacecraft.

The problem to be solved boils down to finding a limited control, limited on magnitudes and rates,

such that the outputs of system (1), 𝑦(𝑡), follow, insofar as possible, the outputs of system (2), 𝑦𝑟(𝑡),

even if parameters are not known accurately, and moreover with asymmetrical magnitude constraints.

Mathematically, the purpose is to find 𝑢(𝑡) such that conditions (3) and (4) remain:

Problem to solve: Find a control 𝑢(𝑡) such that:

‖𝑒𝑦(𝑡)‖ = ‖𝑦(𝑡) − 𝑦𝑟(𝑡)‖ ≤ 𝛿 (3)

for any 𝑡 ≥ 𝜏 wherein 𝜏 > 𝑡𝑢𝑜𝑛, and, obligatorily, such that:

𝑢𝑖,𝑚𝑖𝑛 ≤ 𝑢𝑖(𝑡) ≤ 𝑢𝑖,𝑚𝑎𝑥

|�̇�𝑖(𝑡)| ≤ �̇�𝑖,𝑚𝑎𝑥

, 𝑖 = 1,… ,𝑚 (4)

from the time-instant at which the control is turned on, 𝑡 ≥ 𝑡𝑢𝑜𝑛.

The parameter 𝛿 in inequality (3) denotes the norm of the maximum output error and it should be small as

possible for an appropriate tracking.

3. THE TECHNIQUE PROPOSED

Nonlinear systems (1) and (2) can be decomposed intentionally into the form of a linear part plus a

nonlinear part as indicated in (5) and (6) respectively, wherein 𝐴𝑥 and 𝐴𝑟𝑥𝑟 represent the linear terms,

and 𝜑(𝑥) and 𝜑𝑟(𝑥𝑟) the nonlinear terms:

Controlled system: �̇� = 𝐴𝑥 + 𝐵𝑢 + 𝜑(𝑥)𝑦 = 𝐶𝑥

(5)

Reference system: �̇�𝑟 = 𝐴𝑟𝑥𝑟 + 𝜑𝑟(𝑥𝑟)𝑦𝑟 = 𝐶𝑟𝑥𝑟

(6)

Consider that matrices 𝐴 ∈ ℝ𝑛×𝑛 and 𝐴𝑟 ∈ ℝ𝑛𝑟×𝑛𝑟 are obtained through a classic linearization around

specified states 𝑥∗ and 𝑥𝑟∗ and that functions 𝜑(𝑥) and 𝜑𝑟(𝑥𝑟) are defined by the remaining terms that

complete the global dynamics of each corresponding system. That is:

𝐴 =𝜕𝑓

𝜕𝑥|𝑥 = 𝑥∗

, 𝜑(𝑥) = 𝑓(𝑥) − 𝐴𝑥 (7)

𝐴𝑟 =𝜕𝑓𝑟

𝜕𝑥𝑟|𝑥𝑟 = 𝑥𝑟

∗ , 𝜑𝑟(𝑥𝑟) = 𝑓𝑟(𝑥𝑟) − 𝐴𝑟𝑥𝑟 (8)

Systems (5) and (6) are written without parameter vectors 𝜃 ∈ ℝ𝑝 and 𝜃𝑟 ∈ ℝ𝑝𝑟 for simplification

purposes of the terminology throughout the paper. Nevertheless, be aware that parameters still implicitly

in each system either on matrices 𝐴 and 𝐴𝑟 either on functions 𝜑 and 𝜑𝑟.

Consider now an augmented system comprising the system to be controlled �̇� = 𝐴𝑥 + 𝐵𝑢 + 𝜑(𝑥) and an

auxiliary system, related exclusively with the control vector, governed by differential equation �̇�𝑠 =−𝐿𝑢𝑠 + 𝐿𝑠𝑎𝑡𝑢0

(𝑢𝜀). This last equation results from a generalization of the work presented in (Hippe,

2006) do deal with systems with multiple inputs subject to control constraints, and it is fundamental to

enforce, only in one step, both magnitude and rate constraints. Magnitude constraints are imposed through

the saturation function 𝑠𝑎𝑡𝑢0(𝑢𝜀), given by (9), and rate constraints imposed through the own differential

equation �̇�𝑠 if matrix 𝐿 ∈ ℝ𝑚×𝑚 involve all saturation limits as described in (10):

𝑠𝑎𝑡𝑢0,𝑖(𝑢𝑖) = {

𝑢0,𝑖 , 𝑢𝑖 > 𝑢0,𝑖

𝑢𝑖 , − 𝑢0,𝑖 ≤ 𝑢𝑖 ≤ 𝑢0,𝑖

−𝑢0,𝑖 , 𝑢𝑖 < −𝑢0,𝑖

, 𝑖 = 1,… ,𝑚 (9)

𝐿 = 12⁄

[ �̇�𝑚𝑎𝑥,1/𝑢0,1 0 0 0

0 �̇�𝑚𝑎𝑥,2/𝑢0,2 0 0

0 ⋱ ⋮0 0 … �̇�𝑚𝑎𝑥,𝑚/𝑢0,𝑚]

, 𝑖 = 1,… ,𝑚 (10)

Let the effective control vector, 𝑢 ∈ ℝ𝑚 , be calculated as 𝑢 = 𝑢𝑠 + 𝑢𝑐 wherein 𝑢𝑐 ∈ ℝ𝑚 denotes a

central vector between maximum and minimum saturation limits 𝑢𝑚𝑎𝑥 = [𝑢1,𝑚𝑎𝑥, … , 𝑢𝑚,𝑚𝑎𝑥]𝑇

, 𝑢𝑚𝑖𝑛 =

[𝑢1,𝑚𝑖𝑛, … , 𝑢𝑚,𝑚𝑖𝑛]𝑇

, that is, 𝑢𝑐 = (𝑢𝑚𝑎𝑥 + 𝑢𝑚𝑖𝑛)/2. 𝑢0 = [𝑢0,1, … , 𝑢0,𝑚]𝑇

∈ ℝ𝑚 denotes an auxiliary

vector containing symmetric limits of 𝑢𝑠(𝑡) ∈ ℝ𝑚 and is given by the distance between the vector of

superior limits 𝑢𝑚𝑎𝑥 (or the inferior limits 𝑢𝑚𝑖𝑛 ) and the central vector 𝑢𝑐 , that is, 𝑢0 = 𝑢𝑚𝑎𝑥 − 𝑢𝑐 .

𝑢𝜀 ∈ ℝ𝑚 denotes a virtual control and is based upon it that a controller is designed to stabilize the

augmented system satisfying asymmetrical constraints as specified in (4). The resulting augmented

system has the following form: �̇�𝑎 = 𝐴𝑎𝑥𝑎 + 𝐵𝑎𝑠𝑎𝑡𝑢0(𝑢𝜀) + 𝜑𝑎(𝑥𝑎):

[�̇��̇�𝑠

] = [𝐴 𝐵0𝑚×𝑛 −𝐿

] [𝑥𝑢𝑠

] + [0𝑛×𝑚

𝐿] 𝑠𝑎𝑡𝑢0

(𝑢𝜀) + [𝐵𝑢𝑐 + 𝜑(𝑥)

0𝑚×1] (11)

where 𝑥𝑎 ∈ ℝ𝑛𝑎 denotes the augmented state vector, 𝐴𝑎 ∈ ℝ𝑛𝑎×𝑛𝑎 the augmented state matrix, 𝐵𝑎 ∈ℝ𝑛𝑎×𝑚 the augmented input matrix, 𝜑𝑎 ∈ ℝ𝑛𝑎 a nonlinear function that from now on will be considered

as a disturbance for the augmented system (11), and 𝑛𝑎 = 𝑛 + 𝑚.

Consider now a continuous-time time-invariant system composed by the three equations defined below as

previously adopted by (Bousson & Velosa, 2014b):

�̇�𝑎 = 𝐴𝑎𝑥𝑎 + 𝐵𝑎𝑠𝑎𝑡𝑢0

(𝑢𝜀) + 𝐸𝑤

�̇� = 𝑆𝑤 + 𝑟𝑒 = 𝐶𝑎𝑥𝑎 + 𝐷𝑤

(12)

where the first equation describes the augmented system (11) with the disturbance term 𝜑𝑎 obtained now

through 𝐸𝑤 , the second equation, called exosystem, describes an auxiliary system with sate 𝑤 ∈ ℝ𝑠 ,

𝑠 = 𝑛𝑎 + 𝑛𝑟, that models the disturbance 𝜑𝑎 and the reference signals to be tracked 𝑥𝑟 ∈ ℝ𝑛𝑟, and the

third equation defines the error 𝑒 ∈ ℝ𝑞 between the actual system outputs 𝐶𝑎𝑥𝑎 = 𝑦 and the reference

outputs 𝐷𝑤 = −𝑦𝑟 (also generated by the exosystem).

Let matrices 𝐶𝑎 ∈ ℝ𝑞×𝑛𝑎 , 𝐷 ∈ ℝ𝑞×𝑠 , 𝐸 ∈ ℝ𝑛𝑎×𝑠 by given as 𝐶𝑎 = [𝐶 0𝑞×𝑚] , 𝐷 = [0𝑞×𝑛𝑎 − 𝐶𝑟] ,

𝐸 = [𝐼𝑛𝑎 0𝑛𝑎×𝑛𝑟

], and the exosystem �̇� be described as follows:

�̇� = [�̇�𝑎

�̇�𝑟] = [

0𝑛𝑎×𝑛𝑎0𝑛𝑎×𝑛𝑟

0𝑛𝑟×𝑛𝑎𝐴𝑟

] [𝜑𝑎

𝑥𝑟] + [

[( 𝜕𝑓/𝜕𝑥. �̇� − 𝐴�̇�)𝑇 , 01×𝑚 ]𝑇

𝜑𝑟(𝑥𝑟)] (13)

Theorem 1:

The problem of output regulation via state-feedback for the system (12) is solvable if and only if the

following conditions are true:

i) The pair (𝐴𝑎 , 𝐵𝑎) is stabilizable and 𝐴𝑎 has all its eigenvalues in the closed left half plane.

ii) There exist matrices 𝛱 and 𝛤 such that they solve the so called regulator equations:

𝛱𝑆 = 𝐴𝑎𝛱 + 𝐵𝑎𝛤 + 𝐸

0 = 𝐶𝑎𝛱 + 𝐷 (14)

Under these conditions, the family of linear static state feedback laws given by:

𝑢𝜀 = −𝑅−1𝐵𝑎𝑇𝑃𝜀𝑥𝑎 + (𝐵𝑎

𝑇𝑃𝜀𝛱 + 𝛤)𝑤 + �̃� (15)

where 𝑃𝜀 ∈ ℝ𝑛𝑎×𝑛𝑎 is a symmetric and positive definite matrix (𝑃𝜀 = 𝑃𝜀𝑇 , 𝑃𝜀 > 0), solution of Algebraic

Riccati Equation (ARE):

(𝐴𝑎 + 𝛾𝐼)𝑇𝑃𝜀 + 𝑃𝜀(𝐴𝑎 + 𝛾𝐼) − 𝑃𝜀𝐵𝑎𝑅−1𝐵𝑎𝑇𝑃𝜀 + 𝑄𝜀 = 0 (16)

wherein 𝑄𝜀 ∈ ℝ𝑛𝑎×𝑛𝑎 and 𝑅 ∈ ℝ𝑚×𝑚 are two weighting matrices, both symmetric and positive definite

(𝑄𝜀 = 𝑄𝜀𝑇 , 𝑄𝜀 > 0), (𝑅 = 𝑅𝑇, 𝑅 > 0), 𝐼 ∈ ℝ𝑛𝑎×𝑛𝑎 denotes an identity matrix, 𝛾 ≥ 0 a scalar number

denoting a specified degree of stability, and �̃� = 𝐵𝑎+𝛱𝑟 where 𝐵𝑎

+ denotes the pseudo-inverse matrix of

𝐵𝑎, steers the error 𝑒(𝑡) of system (12) to the origin when 𝑡 → ∞, 𝑙𝑖𝑚𝑡→∞ 𝑒(𝑡) = 0.

A way to solve the regulator equations (14) resorts to a matrix vectorization and can be found in (Bousson

& Velosa, 2014b; Huang, 2004; Saberi, Stoorvogel, & Sannuti, 2011).

To sum up, computing the virtual control 𝑢𝜀(𝑡) by law (15) and initializing the augmented system (11)

with 𝑥(𝑡0) = 𝑥0 and 𝑢𝑠(𝑡0) = −𝑢𝑐 , the auxiliary variable 𝑢𝑠(𝑡) satisfies the magnitude and rate

constraints −𝑢0,𝑖 ≤ 𝑢𝑠,𝑖(𝑡) ≤ 𝑢0,𝑖 , |�̇�𝑠,𝑖(𝑡)| ≤ �̇�𝑠,𝑖,𝑚𝑎𝑥 , and consequently the effective control 𝑢(𝑡)

satisfies 𝑢𝑖,𝑚𝑖𝑛 ≤ 𝑢𝑖(𝑡) ≤ 𝑢𝑖,𝑚𝑎𝑥, |�̇�𝑖(𝑡)| ≤ �̇�𝑖,𝑚𝑎𝑥, with 𝑖 = 1,… ,𝑚. If no control is required to apply on

system (12), the virtual control 𝑢𝜀(𝑡) = −𝑢𝑐 should be maintained. This way, lim𝑡→∞ 𝑢𝑠(𝑡) = −𝑢𝑐 and

lim𝑡→∞ 𝑢(𝑡) = 0.

To end this section one presents a straightforward way to compute the derivatives/rates of the effective

control vector, �̇�(𝑡). Instead of computing them through numerical methods, as example by the method of

the centered finite differences, they can be easily obtained through the time-derivative of 𝑢 = 𝑢𝑠 + 𝑢𝑐, in

what yields:

�̇� = �̇�𝑠 + �̇�𝑐 = −𝐿𝑢𝑠 + 𝐿𝑠𝑎𝑡𝑢0(𝑢𝜀) (17)

4. SIMULATION RESULTS

Hereafter, the technique proposed in the previous section is applied in two aerospace applications to

validate its effectiveness. First, it is applied to supress undesirable and chaotic motions in the attitude of a

spacecraft that may arise due to disturbances present in space, and secondly, applied to control the

position of a spacecraft in a restricted three-body problem, the Earth-Moon-Spacecraft system. In both

applications asymmetric control constraints are considered and the reference outputs are assumed to be

constant.

4.1. Application 1: Spacecraft Attitude Spacecraft attitude refers to the angular orientation of a spacecraft-fixed coordinate frame with respect to

an external reference frame. Assuming that the spacecraft does not contain any moving parts neither

flexible structures, its dynamics can be modelled as a rigid body rotating in space. The attitude of a rigid

body may be described in several ways (direction cosine matrix, Eulerian angles, quaternions, Rodrigues

parameters, Cayley-Klein parameters, Tsiotras-Longuski parameters, etc.), (Shuster, 1993; Tsiotras &

Longuski, 1995). The most common is through the renowned Eulerian angles, which consists in a set of

kinematic equations that relates the attitude angles with the angular velocities of the spacecraft around

each axis, and in another set of dynamical equations that describes the evolution of those angular

velocities. Euler angles are defined with respect to an inertial reference frame by the angle of roll 𝜙, the

angle of pitch 𝜃 and the angle of yaw 𝜓, with the body frame positioned at the center of mass of the

spacecraft. By convention, the 𝑥-axis is placed in the axial direction of the spacecraft pointing in the

direction of flight, the 𝑦-axis points to the right, and the 𝑧-axis points to down, as depicted in figure (1):

Figure 1. Coordinate systems and attitude representation of a rigid-body spacecraft with Euler angles.

Euler angles are convenient particularly when it comes to the “visualization of the attitude”. However,

when it comes to the numerical integration of the differential equations governing the spacecraft

orientation, some singularities occur if one of the angles (depending on the rotation sequence between

frames to obtain the general rotation matrix) is equal to ±90°. By way of example, if the pitch angle is

equal to 𝜃 = 90°, the roll axis 𝑥 becomes parallel to the yaw axis 𝑧′ and cease to be available axes for the

yaw rotation. One degree of freedom is lost - the so called gimbal lock. To overcome such singularities,

quaternions, which are elements of a vector in a four dimensional space that allow to define the rotation

of a rigid body in a three dimensional space, should be employed to describe computationally the attitude

and Euler angles should be used merely to visualize in which state is the spacecraft.

The attitude of a rigid-body spacecraft described in terms of quaternions with respect to an inertial frame

is (Tewari, 2011; Zipfel, 2007):

�̇�𝑥 = 𝑎𝑥𝜔𝑦𝜔𝑧 + 𝑀𝑥/𝐼𝑥𝑥 + 𝑈𝑥/𝐼𝑥𝑥

�̇�𝑦 = 𝑎𝑦𝜔𝑥𝜔𝑧 + 𝑀𝑦/𝐼𝑦𝑦 + 𝑈𝑦/𝐼𝑦𝑦

�̇�𝑧 = 𝑎𝑧𝜔𝑥𝜔𝑦 + 𝑀𝑧/𝐼𝑧𝑧 + 𝑈𝑧/𝐼𝑧𝑧 �̇�0 = (1/2)(−𝜔𝑥𝑞1 − 𝜔𝑦𝑞2 − 𝜔𝑧𝑞3) + 𝑘𝛾𝑞0

�̇�1 = (1/2)(𝜔𝑥𝑞0 + 𝜔𝑧𝑞2 − 𝜔𝑦𝑞3) + 𝑘𝛾𝑞1

�̇�2 = (1/2)(𝜔𝑦𝑞0 − 𝜔𝑧𝑞1 − 𝜔𝑥𝑞3) + 𝑘𝛾𝑞2

�̇�3 = (1/2)(𝜔𝑧𝑞0 + 𝜔𝑦𝑞1 − 𝜔𝑥𝑞2) + 𝑘𝛾𝑞3

(18)

where 𝐼𝑥𝑥, 𝐼𝑦𝑦, 𝐼𝑧𝑧 denote the principal moments of inertia, 𝑎𝑥 = (𝐼𝑦𝑦 − 𝐼𝑧𝑧)/𝐼𝑥𝑥, 𝑎𝑦 = (𝐼𝑧𝑧 − 𝐼𝑥𝑥)/𝐼𝑦𝑦,

𝑎𝑧 = (𝐼𝑥𝑥 − 𝐼𝑦𝑦)/𝐼𝑧𝑧 , 𝑀𝑥 , 𝑀𝑦 , 𝑀𝑧 three perturbing torques existing in space which may change

considerably the spacecraft’s attitude, 𝑈𝑥 , 𝑈𝑦 , 𝑈𝑧 three control torques, and 𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧 the angular

velocities, all with respect to the body axes 𝑥, 𝑦, 𝑧, respectively.

In the second set of differential equations (18), 𝑘𝛾𝑞𝑖, 𝑖 = 0,… ,3, are additional terms to ensure that the

unit norm of the rotation quaternion is maintained even in the presence of rounding errors originated

during the numerical integration. A proven method to maintain the unit norm consists in choosing

𝑘Δ𝑡 ≤ 1, where Δ𝑡 denotes the integration step, and in computing 𝛾 as 𝛾 = 1 − (𝑞02 + 𝑞1

2 + 𝑞22 + 𝑞3

2),

(Zipfel, 2007).

Once obtained the quaternions by numerical integration of (18), the results are easily converted to

physical meaningful quantities as the spacecraft rotates in space. Those are the Euler angles and are

computed at each step as follows:

𝜙 = arctan(2(𝑞2𝑞3 + 𝑞0𝑞1)/(𝑞02 − 𝑞1

2 − 𝑞22 + 𝑞3

2))𝜃 = arcsin(−2(𝑞1𝑞3 − 𝑞0𝑞2))

𝜓 = arctan(2(𝑞1𝑞2 + 𝑞0𝑞3)/(𝑞02 + 𝑞1

2 − 𝑞22 − 𝑞3

2))

(19)

The first and third equations of (19) have singularities at 𝜙 = ±90° and 𝜓 = ±90° respectively.

Nevertheless, this is not problematic because equations (19) are just outputs results calculated off-line to

create an image of the attitude of the spacecraft and therefore can be easily bypassed by programming

around the singularities. What is important is that singularities do not occur inside the differential

equations.

The initialization of the differential equations (18) requires the initial quaternions 𝑞0(0), 𝑞1(0), 𝑞2(0),

𝑞3(0), which are obviously related with the initial attitude of the spacecraft described in terms of Euler

angles 𝜙(0), 𝜃(0), 𝜓(0). The relationship between quaternions and Euler angles is given by expressions

(20):

𝑞0 = cos(𝜓/2) cos(𝜃/2) cos(𝜙/2) + sin(𝜓/2) sin(𝜃/2) sin(𝜙/2)

𝑞1 = cos(𝜓/2) cos(𝜃/2) sin(𝜙/2) − sin(𝜓/2) sin(𝜃/2) cos(𝜙/2)

𝑞2 = cos(𝜓/2) sin(𝜃/2) cos(𝜙/2) + sin(𝜓/2) cos(𝜃/2) sin(𝜙/2)

𝑞3 = sin(𝜓/2) cos(𝜃/2) cos(𝜙/2) − cos(𝜓/2) sin(𝜃/2) sin(𝜙/2)

(20)

If no torques act on the spacecraft, the pattern of rotations existing initially would be maintained

indefinitely. However, experience has shown that, depending on the orbit and spacecraft characteristics,

several different types of disturbing torques are present in space which may change considerably the

spacecraft’s attitude. The total perturbing torque is typically minimum, but its permanence and periodicity

affects the attitude motion in an undesirable way, and, inclusive, can force the spacecraft into a chaotic or

even into a hyperchaotic attitude motion if the matrix of perturbing torques satisfies certain conditions

(Kong, Zhou, & Zou, 2006).

The principal causes that give rise to the perturbing torques are, among others, due to: (1) Aerodynamic

effects; (2) Electromagnetic induction; (3) Solar radiation pressure; (4) Gravity-gradient; (5)

Micrometeorites, (Bousson, 2004; Gerlach, 1965; Kong et al., 2006; Ruiter, Damaren, & Forbes, 2013).

Aerodynamic torques arise due to the Earth’s residual atmosphere and are more significant close to the

Earth, electromagnetic torques arise due to the interaction between the spacecraft’s magnetic field and the

Earth’s magnetic field, solar radiation torques arise due to both electromagnetic radiation and particles

emitted outward the sun, gravity-gradient torques arise due to the small difference in gravitational

attraction from one end to another end of the spacecraft, torques due to micrometeorites arise in regions

such as near the Saturn ring, the Mars- and Jupiter- asteroid belts, and all these perturbing torques have

relations with the angular velocities 𝜔𝑥 , 𝜔𝑦, 𝜔𝑧 and with the attitude angles 𝜙, 𝜃, 𝜓.

Rewriting the first three differential equations of (18) in the vector notation and considering perturbing

torques 𝑀𝑥 , 𝑀𝑦 , 𝑀𝑧 linearly related with the spacecraft angular velocities 𝜔 = [𝜔𝑥 𝜔𝑦 𝜔𝑧]𝑇

, that is,

�̇� = [𝑎𝑥𝜔𝑦𝜔𝑧 𝑎𝑦𝜔𝑥𝜔𝑧 𝑎𝑧𝜔𝑥𝜔𝑦]𝑇

+ 𝐼−1𝑀𝜔 + 𝐼−1𝑢, wherein 𝐼 = diag([𝐼𝑥𝑥 𝐼𝑦𝑦 𝐼𝑧𝑧]) denotes the inertia

matrix, 𝑢 = [𝑈𝑥 𝑈𝑦 𝑈𝑧]𝑇

the vector of control torques, and 𝑀 the matrix of total perturbing torques, one

has:

[

�̇�𝑥

�̇�𝑦

�̇�𝑧

] = [

𝑎𝑥𝜔𝑦𝜔𝑧

𝑎𝑦𝜔𝑥𝜔𝑧

𝑎𝑧𝜔𝑥𝜔𝑦

] + [

𝑚𝑥1/𝐼𝑥 𝑚𝑥2/𝐼𝑥 𝑚𝑥3/𝐼𝑥𝑚𝑦1/𝐼𝑦 𝑚𝑦2/𝐼𝑦 𝑚𝑦3/𝐼𝑦𝑚𝑧1/𝐼𝑧 𝑚𝑧2/𝐼𝑧 𝑚𝑧3/𝐼𝑧

] [

𝜔𝑥

𝜔𝑦

𝜔𝑧

] + [

1/𝐼𝑥 0 00 1/𝐼𝑦 0

0 0 1/𝐼𝑧

] [

𝑈𝑥

𝑈𝑦

𝑈𝑧

] (21)

The matrix of total perturbing torques depends on the own characteristics of the spacecraft as well as on

the characteristics of its orbit/trajectory and is given by 𝑀 = 𝑀𝑎 + 𝑀𝑒 + 𝑀𝑠 + 𝑀𝑔 + 𝑀𝑚 + ⋯, where 𝑀𝑎

denotes the aerodynamic torque matrix, 𝑀𝑒 the electromagnetic torque matrix, 𝑀𝑠 the solar radiation

torque matrix, 𝑀𝑔 the gravitational torque matrix, 𝑀𝑚 the micrometeorite torque matrix, and … additional

torque matrices. Note that if one takes 𝐼𝑥𝑥 = 2𝐼𝑦𝑦 = 2𝐼𝑧𝑧 ⟹ (𝑎𝑥 , 𝑎𝑦, 𝑎𝑧) = (0,−1,1) , which is a

common practice when designing a symmetrical spacecraft with respect to the axial axis to stabilize the

spinning motion around the 𝑥-axis and takes the matrix of total perturbing torques 𝑀 such that:

𝐼−1𝑀 = [−10 10 0

28 −1 00 0 −8/3

] (22)

it can be easily confirmed that the perturbed and uncontrolled system (21) becomes absolutely equal to

the generalized Lorenz system (23) which exhibits a chaotic attractor on its phase-space and has at least

one positive Lyapunov exponent 𝜆 = (0.90, 0, −14.57), (Lorenz, 1963):

[

�̇�𝑥

�̇�𝑦

�̇�𝑧

] = [−10 10 0

28 −1 00 0 −8/3

] [

𝜔𝑥

𝜔𝑦

𝜔𝑧

] + [

0−𝜔𝑥𝜔𝑧

𝜔𝑥𝜔𝑦

] (23)

Once the technique proposed in section (3) requires the outputs of the system written as a linear

combination of the state variables (𝑦 = 𝐶𝑥 ) and the desired outputs are on the contrary nonlinear

functions (attitude angles 𝜙, 𝜃, 𝜓 are nonlinear functions of quaternions), one can consider, when

designing the controller, that quaternions are the real outputs of the system and one can calculate the

reference outputs apart, through the reference attitude angles as indicated in (20). That is, consider

𝑦 = 𝑞 = 𝐶[𝜔 𝑞]𝑇 ⇒ 𝐶 = [04×3 𝐼4] ∈ ℝ4×7 . Observing the system (21) the control input matrix is

𝐵 = 𝐼−1.

The augmented system as required in (11), �̇�𝑎 = 𝐴𝑎𝑥𝑎 + 𝐵𝑎𝑠𝑎𝑡𝑢0(𝑢𝜀) + 𝜑𝑎(𝑥𝑎), comes in the form (24)

and if one computes 𝐴 for example at point 𝑥∗ = [𝜔∗, 𝑞∗]𝑇 with 𝜔∗ = [𝜔𝑥 , 𝜔𝑦 , 𝜔𝑧] = [30, 30, 30] °/s and

𝑞∗ = [𝑞0, 𝑞1, 𝑞2, 𝑞3] such that (𝜙, 𝜃, 𝜓) = (0, 0, 0)°, it can be perfectly controlled through the technique

proposed in section (3), i.e., with control law (15), because the pair (𝐴𝑎 , 𝐵𝑎) is stabilizable, 𝑟𝑎𝑛𝑘(𝑀𝑐) =10 = 𝑛𝑎, and 𝐴𝑎 has all its eigenvalues in the closed left half plane. In this case in particular the point 𝑥∗

cannot be the origin because at that point the resulting augmented system is not stabilizable.

�̇�𝑎 = [�̇��̇�𝑠

] = [𝐴 𝐵03×7 −𝐿

] [𝑥𝑢𝑠

] + [07×3

𝐿] 𝑠𝑎𝑡𝑢0

(𝑢𝜀) + [𝐵𝑢𝑐 + 𝜑(𝑥)

03×1] (24)

�̇� = [�̇�𝑎

�̇�𝑟] = [

010×10 010×4

04×10 04×4] [

𝜑𝑎

𝑥𝑟] + [

[( 𝜕𝑓/𝜕𝑥. �̇� − 𝐴�̇�)𝑇 , 01×3 ]𝑇

04×1] (25)

Consider that the purpose of the control is to stabilize the attitude of the spacecraft around the origin (𝜙, 𝜃, 𝜓) = (0, 0, 0)° when the spacecraft is perturbed according with disturbance (22). The reference

outputs to be tracked are thus constant and equal to 𝑦𝑟(𝑡) = [𝑞0, 𝑞1, 𝑞2, 𝑞3]𝑇 = [1, 0, 0, 0]𝑇.

Considering a four-dimensional reference system given by �̇�𝑟 = 𝐴𝑟𝑥𝑟 + 𝜑𝑟(𝑥𝑟) as required in (6), the

reference outputs 𝑦𝑟 = 𝐶𝑟𝑥𝑟 are easily obtained putting 𝐴𝑟 = 04×4 and 𝜑𝑟(𝑥𝑟) = 04×1 , 𝐶𝑟 = 𝐼4 and

initializing it with 𝑥𝑟,0 = [𝑞0, 𝑞1, 𝑞2, 𝑞3]𝑇 = [1, 0, 0, 0]𝑇. This way one has 𝑦𝑟(𝑡) = 𝑥𝑟(𝑡) = 𝑥𝑟,0 because

�̇�𝑟 = 0.

Simulation:

In what follows the augmented systems (24) and (25) are solved simultaneously through the RK-Butcher

method between 𝑡0 = 0 and 𝑡𝑓 = 15 s, with a step of ∆𝑡 = 0.01 s, and departing from initial conditions

(26). For these initial conditions and for the moments of inertia (𝐼𝑥𝑥, 𝐼𝑦𝑦 , 𝐼𝑧𝑧) = (40, 20, 20) kg.m2 ,

which represents for instance a microsatellite (for a mass < 100 kg, its moment of inertia is usually not

more than 20 kg.m2 (Yang & Sun, 2002)), the attitude of the uncontrolled spacecraft is chaotic and its

angular velocities, governed by the first set of equations (18), exhibit a Lorenz type attractor in its phase-

space as shows figure (2). The parameter 𝑘 required in (18) was defined as being 𝑘 = 1/∆𝑡 and the

parameters assumed for the controller are the ones presented in (27) and the constraints for the controls

(torques and rates) in (28).

Initial conditions:

𝑥0 = [𝜔𝑥, 𝜔𝑦, 𝜔𝑧, 𝑞0, 𝑞1, 𝑞2, 𝑞3]𝑇

with (𝜔𝑥 , 𝜔𝑦, 𝜔𝑧) = (10, 10, 10) °/s

(𝑞0, 𝑞1, 𝑞2, 𝑞3) 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 (𝜙, 𝜃, 𝜓) = (30, 45, 30)°

𝑥𝑟,0 = [𝑞0, 𝑞1, 𝑞2, 𝑞3]𝑇 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 (𝜙, 𝜃, 𝜓) = (0, 0, 0)° , 𝑢𝑠,0 = −𝑢𝑐 , 𝜑𝑎,0 = [

𝐵𝑢𝑐 + 𝜑(𝑥0)03×1

]

(26)

Controller parameters:

휀 = 100 , 𝛾 = 0.4 , 𝑅 = 𝐼3 , 𝑄 = 휀. 𝐼10 (27)

Actuator constraints:

−200 ≤ 𝑢1 ≤ 150

−150 ≤ 𝑢2 ≤ 150

−150 ≤ 𝑢3 ≤ 200

,

|�̇�1| ≤ 3000

|�̇�2| ≤ 2500

|�̇�3| ≤ 2000

(28)

Figure 2. Lorenz type attractor: phase-space of the angular velocities of the uncontrolled spacecraft.

Figures (3) and (4) represent the attitude of the spacecraft described in terms of Euler angles. Figure (3)

showing the time evolution of the eulerian angles (𝜙, 𝜃, 𝜓) together with the reference angles, and figure

(4) the spacecraft angular velocities (𝜔𝑥 , 𝜔𝑦, 𝜔𝑧) with respect to the body axes 𝑥, 𝑦, 𝑧 respectively. The

control torques applied in each axis to stabilize the spacecraft around the origin are shown in figure (5)

and the respective rates in figure (6). The control is turned on from the scratch, at 𝑡𝑢𝑜𝑛= 0 s.

-1500 -1000 -500 0 500 1000 1500-2000

0

2000

0

500

1000

1500

2000

2500

3000

y [º/s]

Phase Space: x vs

y vs

z

x [º/s]

z [

º/s]

Figure 3. Euler angles of the spacecraft. Figure 4. Angular velocities of the spacecraft.

Figure 5. Control torques applied to the spacecraft. Figure 6. Rates of the control torques.

4.2. Application 2: Spacecraft Position - PCR3BP The restricted three-body problem refers to a gravitational system comprising three bodies wherein one of

the bodies, assumed to be of negligible mass in respect to each of the other two, moves in the surrounding

space due to the gravitational forces exerted by the two primary bodies. From the viewpoint of space

mission design, the problem is extremely relevant and studied very often when it comes as an example to

the orbital transfer of a spacecraft between a planet and a moon. It is well-know that although the motion

of the two primary bodies is known, the motion of the third one is not, and cannot be predicted unless

numerically because the problem has no general analytical solution (Diacu, 1996). Moreover, an

0 5 10 15-20

0

20

40

60

t [s]

, , vs t

[º]

ref

[º]

0 5 10 15-20

0

20

40

60

t [s]

[º]

ref

[º]

0 5 10 15-20

0

20

40

60

t [s]

[º]

ref

[º]

0 5 10 15-20

-10

0

10

20

t [s]

x [

º/s]

x,

y,

z vs t

0 5 10 15-40

-20

0

20

40

t [s]

y [

º/s]

0 5 10 15-10

-5

0

5

10

t [s]

z [

º/s]

0 5 10 15-200

-100

0

100

200

t [s]

Ux [

N.m

]

Real Control Torques: Ux, U

y, U

z vs t

0 5 10 15-200

-100

0

100

200

t [s]

Uy [

N.m

]

0 5 10 15-15

-10

-5

0

5

t [s]

Uz [

N.m

]

0 5 10 15-2000

-1000

0

1000

t [s]

Ux d

ot

[N.m

/s]

Rates of Real Control Torques: Ux dot, U

y dot, U

z dot vs t

0 5 10 15-2000

-1000

0

1000

t [s]

Uy d

ot

[N.m

/s]

0 5 10 15-50

0

50

100

150

t [s]

Uz d

ot

[N.m

/s]

interesting phenomenon occurs: for some initial conditions, which actually may be for some points of the

trajectory, the trajectory of the spacecraft becomes chaotic. Some researchers resort thus to the high

sensitivity to the initial conditions to perform orbital manoeuvres with a minimum expenditure of fuel,

which has been also reported to be lower than the fuel required by the classic Hohmann transfer (Bollt &

Meiss, 1995; Macau & Grebogi, 2006).

Following references (Koon, Lo, Marsden, & Ross, 2001; Mingotti, Topputo, & Bernelli-Zazzera, 2009),

the essential dynamics of the three-dimensional system can be captured by a two-dimensional model

given that the Earth’s and Moon’s mean orbital eccentricities are 0.017 and 0.055, respectively, and the

inclination of the Moon’s orbit relative to the Earth’s orbit is 5.15°, (low values). A standard two-

dimensionless model of the restricted three-body problem is achieved considering a barycentric

counterclockwise rotating frame with angular velocity 𝜔 set to 1, with distance between Earth and Moon

𝑙∗ and the sum of their masses 𝑚∗ = 𝑚𝐸 + 𝑚𝑀 all set to 1, and with a characteristic time 𝑡∗ defined in

such a way the dimensionless universal gravitational constant 𝐺∗ equals also to 1, 𝑡∗ = (𝑙∗3/(𝐺𝑚∗))1/2.

On this frame, Earth and Moon have masses 𝑚𝐸 = 1 − 𝜇 and 𝑚𝑀 = 𝜇, and are located respectively at (𝑥𝐸 , 𝑦𝐸) = (−𝜇, 0) and (𝑥𝑀 , 𝑦𝑀) = (1 − 𝜇, 0), as depicted in figure (7). The parameter 𝜇 ∈ [0,1] is the

mass ratio of the restricted three-body problem which is 𝜇 = 𝑚𝑀/𝑚∗.

Figure 7. Geometry of the Planar Circular Restricted Three-Body Problem (PCR3BP) with normalized

quantities.

The mathematical model of the Earth-Moon-Spacecraft system with dimensionless units is (Caillau,

Daoud, & Gergaud, 2012; No, Lee, Jeon, Lee, & Kim, 2012):

�̈� = 𝑥 + 2�̇� −1−𝜇

𝑟𝐸3 (𝑥 + 𝜇) −

𝜇

𝑟𝑀3 (𝑥 − 1 + 𝜇)

�̈� = 𝑦 − 2�̇� −1−𝜇

𝑟𝐸3 𝑦 −

𝜇

𝑟𝑀3 𝑦

(29)

where (𝑥, 𝑦), (�̇�, �̇�), (�̈�, �̈�) denote the spacecraft position, velocity and acceleration, respectively, and 𝑟𝐸

and 𝑟𝑀 the distances between the spacecraft and the Earth and Moon, respectively, 𝑟𝐸 = √(𝑥 + 𝜇)2 + 𝑦2,

𝑟𝑀 = √(𝑥 − 1 + 𝜇)2 + 𝑦2.

Considering the variables transformation: 𝑥1 = 𝑥, 𝑥2 = 𝑦, 𝑥3 = �̇�, 𝑥4 = �̇�, system (29) reduces to the

form of ordinary differential equations:

�̇�1 = 𝑥3

�̇�2 = 𝑥4

�̇�3 = 𝑥1 + 2𝑥4 −1−𝜇

𝑟𝐸3 (𝑥1 + 𝜇) −

𝜇

𝑟𝑀3 (𝑥1 − 1 + 𝜇) + 𝑢1

�̇�4 = 𝑥2 − 2𝑥3 −1−𝜇

𝑟𝐸3 𝑥2 −

𝜇

𝑟𝑀3 𝑥2 + 𝑢2

(30)

which in turn can be written as �̇� = 𝐴𝑥 + 𝐵𝑢 + 𝜑(𝑥) with 𝜑(𝑥) = 𝑓(𝑥) − 𝐴𝑥 and 𝐴 = 𝜕𝑓/𝜕𝑥. In the last

two equations of system (30), 𝑢1 and 𝑢2 are two additional control variables corresponding to the control

accelerations of the spacecraft in the direction of 𝑥 and 𝑦, respectively. The resulting control matrix 𝐵 is:

𝐵 = [

0 00 01 00 1

] ∈ ℝ4×2 (31)

System (30) has five equilibrium points, typically labelled in orbital mechanics as the five libration or

Lagrange points, and their locations are computed by solving the uncontrolled system (30), that is, with

𝑢 = 0 therein, �̇� = 𝑓(𝑥, 𝑢) = 0. Three of the points are collinear with the 𝑥-axis and the remaining two

located in such a way they form symmetric equilateral triangles with the main bodies.

Hereafter one uses the control approach proposed in section (3) to perform an orbital manoeuvre

considering the Earth-Moon-Spacecraft system. The purpose is to steer the spacecraft to one of the

Lagrangian points assuming that the spacecraft is already in a chaotic motion. Consider for this purpose

the point 𝐿1(𝑥, 𝑦) which is located between Earth and Moon at 𝐿1(0.8369, 0).

Since 𝑥3 = �̇�1 and 𝑥4 = �̇�2, let 𝑥1 and 𝑥2 be the outputs of the system (30). The resulting output matrix 𝐶

is:

𝑦 = [𝑥1

𝑥2] = 𝐶𝑥 ⇒ 𝐶 = [

1 0 0 00 1 0 0

] ∈ ℝ2×4 (32)

The augmented system as required in (11), �̇�𝑎 = 𝐴𝑎𝑥𝑎 + 𝐵𝑎𝑠𝑎𝑡𝑢0(𝑢𝜀) + 𝜑𝑎(𝑥𝑎), comes in the form (33)

and if one computes 𝐴 at point 𝑥∗ = [𝑥1, 𝑥2, 𝑥3, 𝑥4]𝑇 = [0.8369, 0, 0, 0]𝑇, it can be perfectly controlled

through the technique proposed in section (3), i.e., with control law (15), because the pair (𝐴𝑎 , 𝐵𝑎) is

stabilizable, 𝑟𝑎𝑛𝑘(𝑀𝑐) = 6 = 𝑛𝑎, and 𝐴𝑎 has all its eigenvalues in the closed left half plane.

�̇�𝑎 = [�̇��̇�𝑠

] = [𝐴 𝐵02×4 −𝐿

] [𝑥𝑢𝑠

] + [04×2

𝐿] 𝑠𝑎𝑡𝑢0

(𝑢𝜀) + [𝐵𝑢𝑐 + 𝜑(𝑥)

02×1] (33)

�̇� = [�̇�𝑎

�̇�𝑟] = [

06×6 06×2

02×6 02×2] [

𝜑𝑎

𝑥𝑟] + [

[( 𝜕𝑓/𝜕𝑥. �̇� − 𝐴�̇�)𝑇 , 01×2 ]𝑇

02×1] (34)

Being 𝐿1 the final destination of the spacecraft, the reference outputs to be tracked are constant and equal

to 𝑦𝑟(𝑡) = [𝑥1, 𝑥2]𝑇 = [0.8369, 0]𝑇 . Considering a bi-dimensional reference system given by �̇�𝑟 =

𝐴𝑟𝑥𝑟 + 𝜑𝑟(𝑥𝑟) as required in (6), the reference outputs 𝑦𝑟 = 𝐶𝑟𝑥𝑟 are easily obtained putting 𝐴𝑟 = 02×2

and 𝜑𝑟(𝑥𝑟) = 02×1 , 𝐶𝑟 = 𝐼2 , and initializing it with 𝑥𝑟,0 = [𝑥1, 𝑥2]𝑇 = [0.8369, 0]𝑇 . This way once

�̇�𝑟 = 0 one has 𝑦𝑟(𝑡) = 𝑥𝑟(𝑡) = 𝑥𝑟,0.

Simulation:

In what follows the augmented systems (33) and (34) are solved simultaneously through the RK-Butcher

method between 𝑡0 = 0 and 𝑡𝑓 = 180 tu, with a step of ∆𝑡 = 0.002 tu, and departing from initial

conditions (35). For these initial conditions and parameters of the Earth-Moon-Spacecraft system

specified in table (1), the trajectory of the uncontrolled spacecraft is chaotic in space. The control is then

activated at 𝑡𝑢𝑜𝑛= 150 tu to redirect the spacecraft to the Lagrange point 𝐿1. The parameters assumed

for the controller are the ones presented in (36) and the constraints for the controls (accelerations and

jerks) in (37). Note that the time is nondimensional and therefore is presented in tu (time units). When

calculating the real time, 𝑡 should be multiplied by 𝑡∗, 1tu = 4.342 days, and the same should be done

for the nondimensional distances, velocities and accelerations, 1du = 384400 km , 1vu = du/tu and

1acu = du/tu2.

Parameter Value

Earth mass: 𝑚𝐸 5.972 × 1024 kg

Moon mass: 𝑚𝑀 7.346 × 1022 kg

Mass parameter: 𝜇 0.01215

Average Earth-Moon distance: 𝑙∗ 384400 km

Characteristic time: 𝑡∗ 3.752 × 105 s , 4.342 days

Gravitational constant: 𝐺 6.674 × 10−11 Nm2kg−2

Table 1. Parameters of the Earth-Moon system.

Initial conditions:

𝑥0 = [𝑥1, 𝑥2, 𝑥3, 𝑥4]𝑇 = [0.8369, 0, −0.0976, 0]𝑇 , 𝑥𝑟,0 = [𝑥1, 𝑥2]

𝑇 = [0.8369, 0]𝑇

𝑢𝑠,0 = −𝑢𝑐 , 𝜑𝑎,0 = [𝐵𝑢𝑐 + 𝜑(𝑥0)

02×1]

(35)

Controller parameters:

휀 = 1.0 , 𝛾 = 1.5 , 𝑅 = 𝐼2 , 𝑄 = 휀. 𝐼6 (36)

Actuator constraints:

−0.3 ≤ 𝑢1 ≤ 1.8

−2.0 ≤ 𝑢2 ≤ 0.2 ,

|�̇�1| ≤ 180

|�̇�2| ≤ 200 (37)

Figures (8) and (9) show respectively the position and velocity of the spacecraft in separated phase-planes

and figures (10) and (11) the respective time evolutions. The control accelerations and the respective rates

(jerks) needed in each direction of the barycentric rotating frame to steer the spacecraft to the libration

point 𝐿1 are shown in figures (12) and (13), respectively. The (blue) dashed lines indicate the behaviour

of the spacecraft before the control is activated, the (red) solid lines the behaviour with the control

activated, for 𝑡 ≥ 𝑡𝑢𝑜𝑛= 150 tu, and all plots are presented in non-dimensional units. Figures (8) and

(10) represent also the system outputs - the position of the spacecraft.

Figure 8. Position of the spacecraft (Phase-space). Figure 9. Velocity of the spacecraft (Phase-space).

Figure 10. Position of the spacecraft (Time history). Figure 11. Velocity of the spacecraft (Time history).

Figure 12. Control accelerations of the spacecraft. Figure 13. Control jerks of the spacecraft.

-0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x1 = x , [du]

x2 =

y ,

[du]

Posição: x vs y

L1

-3 -2 -1 0 1 2 3-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x3 = u , [vu]

x4 =

v ,

[vu]

Velocidade: u vs v

0 20 40 60 80 100 120 140 160 180-1

0

1

2

t , [tu]

x1 =

x ,

[du]

Posição: (x,y) vs t

0 20 40 60 80 100 120 140 160 180-1

-0.5

0

0.5

1

t , [tu]

x2 =

y ,

[du]

0 20 40 60 80 100 120 140 160 180-4

-2

0

2

4

t , [tu]

x3 =

u ,

[vu]

Velocidade: (u,v) vs t

0 20 40 60 80 100 120 140 160 180-2

0

2

4

t , [tu]

x4 =

v ,

[vu]

149 149.5 150 150.5 151 151.5 152 152.5 153-0.5

0

0.5

1

1.5

2

t , [tu]

u1 =

ax ,

[acu]

Acelerações de Controlo: (ax, a

y) vs t

149 149.5 150 150.5 151 151.5 152 152.5 153-2

-1.5

-1

-0.5

0

0.5

t , [tu]

u2 =

ay ,

[acu]

149 149.5 150 150.5 151 151.5 152 152.5 153-50

0

50

100

150

200

t , [tu]

u1,d

ot =

ax,d

ot ,

[acu/t

u]

Taxas das Acelerações de Controlo: (ax,dot

, ay,dot

) vs t

149 149.5 150 150.5 151 151.5 152 152.5 153-200

-100

0

100

t , [tu]

u2,d

ot =

ay,d

ot ,

[acu/t

u]

5. FUTURE RESEARCH DIRECTIONS

In the present paper a control technique that forces exclusively the outputs of nonlinear systems to track

specified reference signals even in the presence of disturbances and subject both to magnitude and rate

actuator constraints is proposed. Nevertheless, although the purpose has been successfully achieved, there

are two subjects, stated below, that depending on the control application may be improved in a future

work. Note that the model that generates the reference signals must be known. The approach proposed

solves thus the problem when the reference signals are known, but when it comes to track reference

signals without knowledge about its mathematical model even if some characteristics are known, another

solution should be formulated. The authors intend to find a sophisticated and elegant solution for such

particular case based on the present work and on the work previously developed by the same authors

(Bousson & Velosa, 2014a). The other point that may also be improved is the fact that the constraints on

the controls considered herein are asymmetric uniquely on the magnitudes. Rate constraints are

symmetric. It is worth noting that although the approach proposed is applicable to a wide range of control

applications subject to actuator constraints, there are, nevertheless, some very specific applications

wherein asymmetric rate constraints must to be taken also into account.

6. DISCUSSION AND CONCLUSION

The paper concerned aimed to propose a technique for the control of real-world dynamical systems in the

sense that the technique designed takes into account actuator constraints. The paper focuses mainly on the

control of nonlinear systems that are liable to exhibit extremely complex behaviours, like chaos, and that

must to comply, more specifically, with magnitude and rate saturation limits. Recent works have been

already presented by same authors, however, given some highly demanding applications as in the

aerospace/aeronautical engineering, there are some specific systems that require asymmetric saturation

limits, and in that sense the paper contributes with an ingenious solution. Note that however the technique

proposed is applicable also for systems that require symmetric input constraints because, as well-known,

symmetric constraints are a particular case of asymmetric constraints.

One of the concerns considered since the beginning was also the control but exclusively of the outputs

of the system instead of the entire states. This way, the approach may be applicable also in problems of

output tracking covering therefore a larger field of applications.

The technique is formulated based on a decomposition of the nonlinear system into a linear part plus a

nonlinear part and resorts to an auxiliary equation to force the actuators to satisfy magnitude and rate

constraints. Both constraints are achieved by the introduction of a single nonlinearity on the system, a

saturation function on the auxiliary equation, and the asymmetric constraints are achieved through a shift

of virtual symmetric constraints. Then, it is formed an augmented system and a control law is designed to

stabilize this augmented system through and solely through its linear part, whereas the remaining

nonlinear terms are seen as bounded and known disturbances.

The control law is designed resorting to an exosystem who generates simultaneously the reference

outputs to be tracked and the disturbances of the augmented system. Due to this, the technique provides a

degree of robustness once the augmented system is stabilized even in the presence of disturbances. These

disturbances are composed by the remaining nonlinear terms but supports also additional perturbing terms

as process noises and/or extra perturbing terms due to parameter uncertainties. The control law is static,

and in that sense, it requires a low computational power once the parameters of the controller may be

computed offline, it is easy to implement, and does not require finding any appropriate Lyapunov

function, as opposed to other control techniques, which is cumbersome to obtain in most cases.

A parameter of the controller, 휀, allows the designer to find a compromise between the output error

and the control effort. The greater is its value, smaller is the output error, and greater is unsurprisingly the

control effort. Another parameter, 𝛾, allows to shift the eigenvalues of the closed-loop matrix to the left in

the direction of the real axis in the complex plane. This parameter helps to ensure that the closed-loop

matrix of the augmented system is stable. Nevertheless, an attention should be taken into account.

Depending on the values of 휀 and 𝛾, the solution of the Algebraic Riccati Equation (ARE) may not exist

or else it may not be unique. On the other hand, if these parameters are too high, the system may not be

stable because the control effort is high and the control must satisfy at same time the required magnitude

and rate constraints. To help circumvent the problem, another parameter, a diagonal weighting matrix 𝑅,

can be adjusted such that the ARE equation has a unique solution and such that the control effort is lower.

The greater are the elements of this matrix, the smaller are the magnitudes of the control variables.

Numerical simulations are presented to validate the technique proposed. It is applied to suppress

undesirable chaotic motions that may arise in aerospace systems, and two very interesting applications are

considered: the control of the attitude of a spacecraft, and the control of the position of a spacecraft in a

restricted three-body problem (Earth-Moon-Spacecraft system). In the first application, a chaotic attitude

motion appears when possible perturbing torques with particular characteristics are present in space, and

the purpose is, naturally, to suppress the chaotic motion by stabilizing the system around a given

equilibrium state. In the second application, the position of a spacecraft becomes chaotic due to the effect

of gravitational fields exerted by Earth and Moon. The control is then applied to steer the spacecraft to a

specified point in space, one of the Lagrangian points in case of the simulation presented.

The results confirm the effectiveness and robustness of the technique proposed. The attitude of the

spacecraft is stabilized around the origin even in the presence of perturbing torques which in the absence

of control would trigger a chaotic dynamics, and the spacecraft is steered to the Lagrange point 𝐿1 ,

maintaining, in both cases, the output error stable in the sense of Lyapunov, and satisfying the required

magnitude and rate control constraints.

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