suppression of chaotic modes in spacecraft with asymmetric ......1. introduction very recently there...
TRANSCRIPT
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 Covilhรฃ, 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 โparticular characteristicsโ 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, ๐-Synthesis Control, ๐ป2/๐ปโ),
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 (LMIโs), are the most cited techniques available
in literature (Bousson & Velosa, 2014a). However, all techniques proposed present, that is authorsโ
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: ๏ฟฝฬ๏ฟฝ = ๐(๐ฅ, ๐) + ๐ต๐ข๐ฆ = ๐ถ๐ฅ
(1)
Reference system: ๏ฟฝฬ๏ฟฝ๐ = ๐๐(๐ฅ๐, ๐๐)๐ฆ๐ = ๐ถ๐๐ฅ๐
(2)
where ๐ฅ โ โ๐, ๐ฅ๐ โ โ๐๐ represent the state vectors, ๐ โ โ๐, ๐๐ โ โ๐๐ vectors of parameters, ๐ฆ, ๐ฆ๐ โ โ๐
the outputs of each system (vectors with same dimension), ๐ข โ โ๐ the control input vector, ๐ถ โ โ๐ร๐,
๐ถ๐ โ โ๐ร๐๐ the output matrices, ๐ต โ โ๐ร๐ the input matrix, ๐, ๐๐ two nonlinear functions, and ๏ฟฝฬ๏ฟฝ = ๐/๐๐ก.
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 โ๐ข(๐ก)โโ โค ๐ and
โ๏ฟฝฬ๏ฟฝ(๐ก)โโ โค ๐ , with ๐ โฅ 0 and ๐ โฅ 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 [โ25, 35]ยฐ (asymmetrical) instead of [โ30, 30]ยฐ (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), ๐ฆ(๐ก), follow, insofar as possible, the outputs of system (2), ๐ฆ๐(๐ก),
even if parameters are not known accurately, and moreover with asymmetrical magnitude constraints.
Mathematically, the purpose is to find ๐ข(๐ก) such that conditions (3) and (4) remain:
Problem to solve: Find a control ๐ข(๐ก) such that:
โ๐๐ฆ(๐ก)โ = โ๐ฆ(๐ก) โ ๐ฆ๐(๐ก)โ โค ๐ฟ (3)
for any ๐ก โฅ ๐ wherein ๐ > ๐ก๐ข๐๐, and, obligatorily, such that:
๐ข๐,๐๐๐ โค ๐ข๐(๐ก) โค ๐ข๐,๐๐๐ฅ
|๏ฟฝฬ๏ฟฝ๐(๐ก)| โค ๏ฟฝฬ๏ฟฝ๐,๐๐๐ฅ
, ๐ = 1,โฆ ,๐ (4)
from the time-instant at which the control is turned on, ๐ก โฅ ๐ก๐ข๐๐.
The parameter ๐ฟ 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 ๐ด๐ฅ and ๐ด๐๐ฅ๐ represent the linear terms,
and ๐(๐ฅ) and ๐๐(๐ฅ๐) the nonlinear terms:
Controlled system: ๏ฟฝฬ๏ฟฝ = ๐ด๐ฅ + ๐ต๐ข + ๐(๐ฅ)๐ฆ = ๐ถ๐ฅ
(5)
Reference system: ๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐๐(๐ฅ๐)๐ฆ๐ = ๐ถ๐๐ฅ๐
(6)
Consider that matrices ๐ด โ โ๐ร๐ and ๐ด๐ โ โ๐๐ร๐๐ are obtained through a classic linearization around
specified states ๐ฅโ and ๐ฅ๐โ and that functions ๐(๐ฅ) and ๐๐(๐ฅ๐) are defined by the remaining terms that
complete the global dynamics of each corresponding system. That is:
๐ด =๐๐
๐๐ฅ|๐ฅ = ๐ฅโ
, ๐(๐ฅ) = ๐(๐ฅ) โ ๐ด๐ฅ (7)
๐ด๐ =๐๐๐
๐๐ฅ๐|๐ฅ๐ = ๐ฅ๐
โ , ๐๐(๐ฅ๐) = ๐๐(๐ฅ๐) โ ๐ด๐๐ฅ๐ (8)
Systems (5) and (6) are written without parameter vectors ๐ โ โ๐ and ๐๐ โ โ๐๐ for simplification
purposes of the terminology throughout the paper. Nevertheless, be aware that parameters still implicitly
in each system either on matrices ๐ด and ๐ด๐ either on functions ๐ and ๐๐.
Consider now an augmented system comprising the system to be controlled ๏ฟฝฬ๏ฟฝ = ๐ด๐ฅ + ๐ต๐ข + ๐(๐ฅ) and an
auxiliary system, related exclusively with the control vector, governed by differential equation ๏ฟฝฬ๏ฟฝ๐ =โ๐ฟ๐ข๐ + ๐ฟ๐ ๐๐ก๐ข0
(๐ข๐). 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 ๐ ๐๐ก๐ข0(๐ข๐), given by (9), and rate constraints imposed through the own differential
equation ๏ฟฝฬ๏ฟฝ๐ if matrix ๐ฟ โ โ๐ร๐ involve all saturation limits as described in (10):
๐ ๐๐ก๐ข0,๐(๐ข๐) = {
๐ข0,๐ , ๐ข๐ > ๐ข0,๐
๐ข๐ , โ ๐ข0,๐ โค ๐ข๐ โค ๐ข0,๐
โ๐ข0,๐ , ๐ข๐ < โ๐ข0,๐
, ๐ = 1,โฆ ,๐ (9)
๐ฟ = 12โ
[ ๏ฟฝฬ๏ฟฝ๐๐๐ฅ,1/๐ข0,1 0 0 0
0 ๏ฟฝฬ๏ฟฝ๐๐๐ฅ,2/๐ข0,2 0 0
0 โฑ โฎ0 0 โฆ ๏ฟฝฬ๏ฟฝ๐๐๐ฅ,๐/๐ข0,๐]
, ๐ = 1,โฆ ,๐ (10)
Let the effective control vector, ๐ข โ โ๐ , be calculated as ๐ข = ๐ข๐ + ๐ข๐ wherein ๐ข๐ โ โ๐ denotes a
central vector between maximum and minimum saturation limits ๐ข๐๐๐ฅ = [๐ข1,๐๐๐ฅ, โฆ , ๐ข๐,๐๐๐ฅ]๐
, ๐ข๐๐๐ =
[๐ข1,๐๐๐, โฆ , ๐ข๐,๐๐๐]๐
, that is, ๐ข๐ = (๐ข๐๐๐ฅ + ๐ข๐๐๐)/2. ๐ข0 = [๐ข0,1, โฆ , ๐ข0,๐]๐
โ โ๐ denotes an auxiliary
vector containing symmetric limits of ๐ข๐ (๐ก) โ โ๐ and is given by the distance between the vector of
superior limits ๐ข๐๐๐ฅ (or the inferior limits ๐ข๐๐๐ ) and the central vector ๐ข๐ , that is, ๐ข0 = ๐ข๐๐๐ฅ โ ๐ข๐ .
๐ข๐ โ โ๐ 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: ๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐ต๐๐ ๐๐ก๐ข0(๐ข๐) + ๐๐(๐ฅ๐):
[๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ๐
] = [๐ด ๐ต0๐ร๐ โ๐ฟ
] [๐ฅ๐ข๐
] + [0๐ร๐
๐ฟ] ๐ ๐๐ก๐ข0
(๐ข๐) + [๐ต๐ข๐ + ๐(๐ฅ)
0๐ร1] (11)
where ๐ฅ๐ โ โ๐๐ denotes the augmented state vector, ๐ด๐ โ โ๐๐ร๐๐ the augmented state matrix, ๐ต๐ โโ๐๐ร๐ the augmented input matrix, ๐๐ โ โ๐๐ a nonlinear function that from now on will be considered
as a disturbance for the augmented system (11), and ๐๐ = ๐ + ๐.
Consider now a continuous-time time-invariant system composed by the three equations defined below as
previously adopted by (Bousson & Velosa, 2014b):
๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐ต๐๐ ๐๐ก๐ข0
(๐ข๐) + ๐ธ๐ค
๏ฟฝฬ๏ฟฝ = ๐๐ค + ๐๐ = ๐ถ๐๐ฅ๐ + ๐ท๐ค
(12)
where the first equation describes the augmented system (11) with the disturbance term ๐๐ obtained now
through ๐ธ๐ค , the second equation, called exosystem, describes an auxiliary system with sate ๐ค โ โ๐ ,
๐ = ๐๐ + ๐๐, that models the disturbance ๐๐ and the reference signals to be tracked ๐ฅ๐ โ โ๐๐, and the
third equation defines the error ๐ โ โ๐ between the actual system outputs ๐ถ๐๐ฅ๐ = ๐ฆ and the reference
outputs ๐ท๐ค = โ๐ฆ๐ (also generated by the exosystem).
Let matrices ๐ถ๐ โ โ๐ร๐๐ , ๐ท โ โ๐ร๐ , ๐ธ โ โ๐๐ร๐ by given as ๐ถ๐ = [๐ถ 0๐ร๐] , ๐ท = [0๐ร๐๐ โ ๐ถ๐] ,
๐ธ = [๐ผ๐๐ 0๐๐ร๐๐
], and the exosystem ๏ฟฝฬ๏ฟฝ be described as follows:
๏ฟฝฬ๏ฟฝ = [๏ฟฝฬ๏ฟฝ๐
๏ฟฝฬ๏ฟฝ๐] = [
0๐๐ร๐๐0๐๐ร๐๐
0๐๐ร๐๐๐ด๐
] [๐๐
๐ฅ๐] + [
[( ๐๐/๐๐ฅ. ๏ฟฝฬ๏ฟฝ โ ๐ด๏ฟฝฬ๏ฟฝ)๐ , 01ร๐ ]๐
๐๐(๐ฅ๐)] (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 (๐ด๐ , ๐ต๐) is stabilizable and ๐ด๐ has all its eigenvalues in the closed left half plane.
ii) There exist matrices ๐ฑ and ๐ค such that they solve the so called regulator equations:
๐ฑ๐ = ๐ด๐๐ฑ + ๐ต๐๐ค + ๐ธ
0 = ๐ถ๐๐ฑ + ๐ท (14)
Under these conditions, the family of linear static state feedback laws given by:
๐ข๐ = โ๐ โ1๐ต๐๐๐๐๐ฅ๐ + (๐ต๐
๐๐๐๐ฑ + ๐ค)๐ค + ๏ฟฝฬ๏ฟฝ (15)
where ๐๐ โ โ๐๐ร๐๐ is a symmetric and positive definite matrix (๐๐ = ๐๐๐ , ๐๐ > 0), solution of Algebraic
Riccati Equation (ARE):
(๐ด๐ + ๐พ๐ผ)๐๐๐ + ๐๐(๐ด๐ + ๐พ๐ผ) โ ๐๐๐ต๐๐ โ1๐ต๐๐๐๐ + ๐๐ = 0 (16)
wherein ๐๐ โ โ๐๐ร๐๐ and ๐ โ โ๐ร๐ are two weighting matrices, both symmetric and positive definite
(๐๐ = ๐๐๐ , ๐๐ > 0), (๐ = ๐ ๐, ๐ > 0), ๐ผ โ โ๐๐ร๐๐ denotes an identity matrix, ๐พ โฅ 0 a scalar number
denoting a specified degree of stability, and ๏ฟฝฬ๏ฟฝ = ๐ต๐+๐ฑ๐ where ๐ต๐
+ denotes the pseudo-inverse matrix of
๐ต๐, steers the error ๐(๐ก) of system (12) to the origin when ๐ก โ โ, ๐๐๐๐กโโ ๐(๐ก) = 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 ๐ข๐(๐ก) by law (15) and initializing the augmented system (11)
with ๐ฅ(๐ก0) = ๐ฅ0 and ๐ข๐ (๐ก0) = โ๐ข๐ , the auxiliary variable ๐ข๐ (๐ก) satisfies the magnitude and rate
constraints โ๐ข0,๐ โค ๐ข๐ ,๐(๐ก) โค ๐ข0,๐ , |๏ฟฝฬ๏ฟฝ๐ ,๐(๐ก)| โค ๏ฟฝฬ๏ฟฝ๐ ,๐,๐๐๐ฅ , and consequently the effective control ๐ข(๐ก)
satisfies ๐ข๐,๐๐๐ โค ๐ข๐(๐ก) โค ๐ข๐,๐๐๐ฅ, |๏ฟฝฬ๏ฟฝ๐(๐ก)| โค ๏ฟฝฬ๏ฟฝ๐,๐๐๐ฅ, with ๐ = 1,โฆ ,๐. If no control is required to apply on
system (12), the virtual control ๐ข๐(๐ก) = โ๐ข๐ should be maintained. This way, lim๐กโโ ๐ข๐ (๐ก) = โ๐ข๐ and
lim๐กโโ ๐ข(๐ก) = 0.
To end this section one presents a straightforward way to compute the derivatives/rates of the effective
control vector, ๏ฟฝฬ๏ฟฝ(๐ก). 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 ๐ข = ๐ข๐ + ๐ข๐, in
what yields:
๏ฟฝฬ๏ฟฝ = ๏ฟฝฬ๏ฟฝ๐ + ๏ฟฝฬ๏ฟฝ๐ = โ๐ฟ๐ข๐ + ๐ฟ๐ ๐๐ก๐ข0(๐ข๐) (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 ๐, the
angle of pitch ๐ and the angle of yaw ๐, with the body frame positioned at the center of mass of the
spacecraft. By convention, the ๐ฅ-axis is placed in the axial direction of the spacecraft pointing in the
direction of flight, the ๐ฆ-axis points to the right, and the ๐ง-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 โvisualization of the attitudeโ. 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 ยฑ90ยฐ. By way of example, if the pitch angle is
equal to ๐ = 90ยฐ, the roll axis ๐ฅ becomes parallel to the yaw axis ๐งโฒ 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):
๏ฟฝฬ๏ฟฝ๐ฅ = ๐๐ฅ๐๐ฆ๐๐ง + ๐๐ฅ/๐ผ๐ฅ๐ฅ + ๐๐ฅ/๐ผ๐ฅ๐ฅ
๏ฟฝฬ๏ฟฝ๐ฆ = ๐๐ฆ๐๐ฅ๐๐ง + ๐๐ฆ/๐ผ๐ฆ๐ฆ + ๐๐ฆ/๐ผ๐ฆ๐ฆ
๏ฟฝฬ๏ฟฝ๐ง = ๐๐ง๐๐ฅ๐๐ฆ + ๐๐ง/๐ผ๐ง๐ง + ๐๐ง/๐ผ๐ง๐ง ๏ฟฝฬ๏ฟฝ0 = (1/2)(โ๐๐ฅ๐1 โ ๐๐ฆ๐2 โ ๐๐ง๐3) + ๐๐พ๐0
๏ฟฝฬ๏ฟฝ1 = (1/2)(๐๐ฅ๐0 + ๐๐ง๐2 โ ๐๐ฆ๐3) + ๐๐พ๐1
๏ฟฝฬ๏ฟฝ2 = (1/2)(๐๐ฆ๐0 โ ๐๐ง๐1 โ ๐๐ฅ๐3) + ๐๐พ๐2
๏ฟฝฬ๏ฟฝ3 = (1/2)(๐๐ง๐0 + ๐๐ฆ๐1 โ ๐๐ฅ๐2) + ๐๐พ๐3
(18)
where ๐ผ๐ฅ๐ฅ, ๐ผ๐ฆ๐ฆ, ๐ผ๐ง๐ง denote the principal moments of inertia, ๐๐ฅ = (๐ผ๐ฆ๐ฆ โ ๐ผ๐ง๐ง)/๐ผ๐ฅ๐ฅ, ๐๐ฆ = (๐ผ๐ง๐ง โ ๐ผ๐ฅ๐ฅ)/๐ผ๐ฆ๐ฆ,
๐๐ง = (๐ผ๐ฅ๐ฅ โ ๐ผ๐ฆ๐ฆ)/๐ผ๐ง๐ง , ๐๐ฅ , ๐๐ฆ , ๐๐ง three perturbing torques existing in space which may change
considerably the spacecraftโs attitude, ๐๐ฅ , ๐๐ฆ , ๐๐ง three control torques, and ๐๐ฅ , ๐๐ฆ , ๐๐ง the angular
velocities, all with respect to the body axes ๐ฅ, ๐ฆ, ๐ง, respectively.
In the second set of differential equations (18), ๐๐พ๐๐, ๐ = 0,โฆ ,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
๐ฮ๐ก โค 1, where ฮ๐ก denotes the integration step, and in computing ๐พ as ๐พ = 1 โ (๐02 + ๐1
2 + ๐22 + ๐3
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:
๐ = arctan(2(๐2๐3 + ๐0๐1)/(๐02 โ ๐1
2 โ ๐22 + ๐3
2))๐ = arcsin(โ2(๐1๐3 โ ๐0๐2))
๐ = arctan(2(๐1๐2 + ๐0๐3)/(๐02 + ๐1
2 โ ๐22 โ ๐3
2))
(19)
The first and third equations of (19) have singularities at ๐ = ยฑ90ยฐ and ๐ = ยฑ90ยฐ 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 ๐0(0), ๐1(0), ๐2(0),
๐3(0), which are obviously related with the initial attitude of the spacecraft described in terms of Euler
angles ๐(0), ๐(0), ๐(0). The relationship between quaternions and Euler angles is given by expressions
(20):
๐0 = cos(๐/2) cos(๐/2) cos(๐/2) + sin(๐/2) sin(๐/2) sin(๐/2)
๐1 = cos(๐/2) cos(๐/2) sin(๐/2) โ sin(๐/2) sin(๐/2) cos(๐/2)
๐2 = cos(๐/2) sin(๐/2) cos(๐/2) + sin(๐/2) cos(๐/2) sin(๐/2)
๐3 = sin(๐/2) cos(๐/2) cos(๐/2) โ cos(๐/2) sin(๐/2) sin(๐/2)
(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
spacecraftโs 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 Earthโs residual atmosphere and are more significant close to the
Earth, electromagnetic torques arise due to the interaction between the spacecraftโs magnetic field and the
Earthโs 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 ๐๐ฅ , ๐๐ฆ, ๐๐ง and with the attitude angles ๐, ๐, ๐.
Rewriting the first three differential equations of (18) in the vector notation and considering perturbing
torques ๐๐ฅ , ๐๐ฆ , ๐๐ง linearly related with the spacecraft angular velocities ๐ = [๐๐ฅ ๐๐ฆ ๐๐ง]๐
, that is,
๏ฟฝฬ๏ฟฝ = [๐๐ฅ๐๐ฆ๐๐ง ๐๐ฆ๐๐ฅ๐๐ง ๐๐ง๐๐ฅ๐๐ฆ]๐
+ ๐ผโ1๐๐ + ๐ผโ1๐ข, wherein ๐ผ = diag([๐ผ๐ฅ๐ฅ ๐ผ๐ฆ๐ฆ ๐ผ๐ง๐ง]) denotes the inertia
matrix, ๐ข = [๐๐ฅ ๐๐ฆ ๐๐ง]๐
the vector of control torques, and ๐ the matrix of total perturbing torques, one
has:
[
๏ฟฝฬ๏ฟฝ๐ฅ
๏ฟฝฬ๏ฟฝ๐ฆ
๏ฟฝฬ๏ฟฝ๐ง
] = [
๐๐ฅ๐๐ฆ๐๐ง
๐๐ฆ๐๐ฅ๐๐ง
๐๐ง๐๐ฅ๐๐ฆ
] + [
๐๐ฅ1/๐ผ๐ฅ ๐๐ฅ2/๐ผ๐ฅ ๐๐ฅ3/๐ผ๐ฅ๐๐ฆ1/๐ผ๐ฆ ๐๐ฆ2/๐ผ๐ฆ ๐๐ฆ3/๐ผ๐ฆ๐๐ง1/๐ผ๐ง ๐๐ง2/๐ผ๐ง ๐๐ง3/๐ผ๐ง
] [
๐๐ฅ
๐๐ฆ
๐๐ง
] + [
1/๐ผ๐ฅ 0 00 1/๐ผ๐ฆ 0
0 0 1/๐ผ๐ง
] [
๐๐ฅ
๐๐ฆ
๐๐ง
] (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 ๐ = ๐๐ + ๐๐ + ๐๐ + ๐๐ + ๐๐ + โฏ, where ๐๐
denotes the aerodynamic torque matrix, ๐๐ the electromagnetic torque matrix, ๐๐ the solar radiation
torque matrix, ๐๐ the gravitational torque matrix, ๐๐ the micrometeorite torque matrix, and โฆ additional
torque matrices. Note that if one takes ๐ผ๐ฅ๐ฅ = 2๐ผ๐ฆ๐ฆ = 2๐ผ๐ง๐ง โน (๐๐ฅ , ๐๐ฆ, ๐๐ง) = (0,โ1,1) , which is a
common practice when designing a symmetrical spacecraft with respect to the axial axis to stabilize the
spinning motion around the ๐ฅ-axis and takes the matrix of total perturbing torques ๐ such that:
๐ผโ1๐ = [โ10 10 0
28 โ1 00 0 โ8/3
] (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 ๐ = (0.90, 0, โ14.57), (Lorenz, 1963):
[
๏ฟฝฬ๏ฟฝ๐ฅ
๏ฟฝฬ๏ฟฝ๐ฆ
๏ฟฝฬ๏ฟฝ๐ง
] = [โ10 10 0
28 โ1 00 0 โ8/3
] [
๐๐ฅ
๐๐ฆ
๐๐ง
] + [
0โ๐๐ฅ๐๐ง
๐๐ฅ๐๐ฆ
] (23)
Once the technique proposed in section (3) requires the outputs of the system written as a linear
combination of the state variables (๐ฆ = ๐ถ๐ฅ ) and the desired outputs are on the contrary nonlinear
functions (attitude angles ๐, ๐, ๐ 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
๐ฆ = ๐ = ๐ถ[๐ ๐]๐ โ ๐ถ = [04ร3 ๐ผ4] โ โ4ร7 . Observing the system (21) the control input matrix is
๐ต = ๐ผโ1.
The augmented system as required in (11), ๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐ต๐๐ ๐๐ก๐ข0(๐ข๐) + ๐๐(๐ฅ๐), comes in the form (24)
and if one computes ๐ด for example at point ๐ฅโ = [๐โ, ๐โ]๐ with ๐โ = [๐๐ฅ , ๐๐ฆ , ๐๐ง] = [30, 30, 30] ยฐ/s and
๐โ = [๐0, ๐1, ๐2, ๐3] such that (๐, ๐, ๐) = (0, 0, 0)ยฐ, it can be perfectly controlled through the technique
proposed in section (3), i.e., with control law (15), because the pair (๐ด๐ , ๐ต๐) is stabilizable, ๐๐๐๐(๐๐) =10 = ๐๐, and ๐ด๐ has all its eigenvalues in the closed left half plane. In this case in particular the point ๐ฅโ
cannot be the origin because at that point the resulting augmented system is not stabilizable.
๏ฟฝฬ๏ฟฝ๐ = [๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ๐
] = [๐ด ๐ต03ร7 โ๐ฟ
] [๐ฅ๐ข๐
] + [07ร3
๐ฟ] ๐ ๐๐ก๐ข0
(๐ข๐) + [๐ต๐ข๐ + ๐(๐ฅ)
03ร1] (24)
๏ฟฝฬ๏ฟฝ = [๏ฟฝฬ๏ฟฝ๐
๏ฟฝฬ๏ฟฝ๐] = [
010ร10 010ร4
04ร10 04ร4] [
๐๐
๐ฅ๐] + [
[( ๐๐/๐๐ฅ. ๏ฟฝฬ๏ฟฝ โ ๐ด๏ฟฝฬ๏ฟฝ)๐ , 01ร3 ]๐
04ร1] (25)
Consider that the purpose of the control is to stabilize the attitude of the spacecraft around the origin (๐, ๐, ๐) = (0, 0, 0)ยฐ when the spacecraft is perturbed according with disturbance (22). The reference
outputs to be tracked are thus constant and equal to ๐ฆ๐(๐ก) = [๐0, ๐1, ๐2, ๐3]๐ = [1, 0, 0, 0]๐.
Considering a four-dimensional reference system given by ๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐๐(๐ฅ๐) as required in (6), the
reference outputs ๐ฆ๐ = ๐ถ๐๐ฅ๐ are easily obtained putting ๐ด๐ = 04ร4 and ๐๐(๐ฅ๐) = 04ร1 , ๐ถ๐ = ๐ผ4 and
initializing it with ๐ฅ๐,0 = [๐0, ๐1, ๐2, ๐3]๐ = [1, 0, 0, 0]๐. This way one has ๐ฆ๐(๐ก) = ๐ฅ๐(๐ก) = ๐ฅ๐,0 because
๏ฟฝฬ๏ฟฝ๐ = 0.
Simulation:
In what follows the augmented systems (24) and (25) are solved simultaneously through the RK-Butcher
method between ๐ก0 = 0 and ๐ก๐ = 15 s, with a step of โ๐ก = 0.01 s, and departing from initial conditions
(26). For these initial conditions and for the moments of inertia (๐ผ๐ฅ๐ฅ, ๐ผ๐ฆ๐ฆ , ๐ผ๐ง๐ง) = (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 ๐ required in (18) was defined as being ๐ = 1/โ๐ก 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:
๐ฅ0 = [๐๐ฅ, ๐๐ฆ, ๐๐ง, ๐0, ๐1, ๐2, ๐3]๐
with (๐๐ฅ , ๐๐ฆ, ๐๐ง) = (10, 10, 10) ยฐ/s
(๐0, ๐1, ๐2, ๐3) ๐ ๐ข๐โ ๐กโ๐๐ก (๐, ๐, ๐) = (30, 45, 30)ยฐ
๐ฅ๐,0 = [๐0, ๐1, ๐2, ๐3]๐ ๐ ๐ข๐โ ๐กโ๐๐ก (๐, ๐, ๐) = (0, 0, 0)ยฐ , ๐ข๐ ,0 = โ๐ข๐ , ๐๐,0 = [
๐ต๐ข๐ + ๐(๐ฅ0)03ร1
]
(26)
Controller parameters:
ํ = 100 , ๐พ = 0.4 , ๐ = ๐ผ3 , ๐ = ํ. ๐ผ10 (27)
Actuator constraints:
โ200 โค ๐ข1 โค 150
โ150 โค ๐ข2 โค 150
โ150 โค ๐ข3 โค 200
,
|๏ฟฝฬ๏ฟฝ1| โค 3000
|๏ฟฝฬ๏ฟฝ2| โค 2500
|๏ฟฝฬ๏ฟฝ3| โค 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 (๐, ๐, ๐) together with the reference angles, and figure
(4) the spacecraft angular velocities (๐๐ฅ , ๐๐ฆ, ๐๐ง) with respect to the body axes ๐ฅ, ๐ฆ, ๐ง 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 ๐ก๐ข๐๐= 0 s.
-1500 -1000 -500 0 500 1000 1500-2000
0
2000
0
500
1000
1500
2000
2500
3000
y [ยบ/s]
Phase Space: x vs
y vs
z
x [ยบ/s]
z [
ยบ/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
[ยบ]
ref
[ยบ]
0 5 10 15-20
0
20
40
60
t [s]
[ยบ]
ref
[ยบ]
0 5 10 15-20
0
20
40
60
t [s]
[ยบ]
ref
[ยบ]
0 5 10 15-20
-10
0
10
20
t [s]
x [
ยบ/s]
x,
y,
z vs t
0 5 10 15-40
-20
0
20
40
t [s]
y [
ยบ/s]
0 5 10 15-10
-5
0
5
10
t [s]
z [
ยบ/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 Earthโs and Moonโs mean orbital eccentricities are 0.017 and 0.055, respectively, and the
inclination of the Moonโs orbit relative to the Earthโs orbit is 5.15ยฐ, (low values). A standard two-
dimensionless model of the restricted three-body problem is achieved considering a barycentric
counterclockwise rotating frame with angular velocity ๐ set to 1, with distance between Earth and Moon
๐โ and the sum of their masses ๐โ = ๐๐ธ + ๐๐ all set to 1, and with a characteristic time ๐กโ defined in
such a way the dimensionless universal gravitational constant ๐บโ equals also to 1, ๐กโ = (๐โ3/(๐บ๐โ))1/2.
On this frame, Earth and Moon have masses ๐๐ธ = 1 โ ๐ and ๐๐ = ๐, and are located respectively at (๐ฅ๐ธ , ๐ฆ๐ธ) = (โ๐, 0) and (๐ฅ๐ , ๐ฆ๐) = (1 โ ๐, 0), as depicted in figure (7). The parameter ๐ โ [0,1] is the
mass ratio of the restricted three-body problem which is ๐ = ๐๐/๐โ.
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):
๏ฟฝฬ๏ฟฝ = ๐ฅ + 2๏ฟฝฬ๏ฟฝ โ1โ๐
๐๐ธ3 (๐ฅ + ๐) โ
๐
๐๐3 (๐ฅ โ 1 + ๐)
๏ฟฝฬ๏ฟฝ = ๐ฆ โ 2๏ฟฝฬ๏ฟฝ โ1โ๐
๐๐ธ3 ๐ฆ โ
๐
๐๐3 ๐ฆ
(29)
where (๐ฅ, ๐ฆ), (๏ฟฝฬ๏ฟฝ, ๏ฟฝฬ๏ฟฝ), (๏ฟฝฬ๏ฟฝ, ๏ฟฝฬ๏ฟฝ) denote the spacecraft position, velocity and acceleration, respectively, and ๐๐ธ
and ๐๐ the distances between the spacecraft and the Earth and Moon, respectively, ๐๐ธ = โ(๐ฅ + ๐)2 + ๐ฆ2,
๐๐ = โ(๐ฅ โ 1 + ๐)2 + ๐ฆ2.
Considering the variables transformation: ๐ฅ1 = ๐ฅ, ๐ฅ2 = ๐ฆ, ๐ฅ3 = ๏ฟฝฬ๏ฟฝ, ๐ฅ4 = ๏ฟฝฬ๏ฟฝ, system (29) reduces to the
form of ordinary differential equations:
๏ฟฝฬ๏ฟฝ1 = ๐ฅ3
๏ฟฝฬ๏ฟฝ2 = ๐ฅ4
๏ฟฝฬ๏ฟฝ3 = ๐ฅ1 + 2๐ฅ4 โ1โ๐
๐๐ธ3 (๐ฅ1 + ๐) โ
๐
๐๐3 (๐ฅ1 โ 1 + ๐) + ๐ข1
๏ฟฝฬ๏ฟฝ4 = ๐ฅ2 โ 2๐ฅ3 โ1โ๐
๐๐ธ3 ๐ฅ2 โ
๐
๐๐3 ๐ฅ2 + ๐ข2
(30)
which in turn can be written as ๏ฟฝฬ๏ฟฝ = ๐ด๐ฅ + ๐ต๐ข + ๐(๐ฅ) with ๐(๐ฅ) = ๐(๐ฅ) โ ๐ด๐ฅ and ๐ด = ๐๐/๐๐ฅ. In the last
two equations of system (30), ๐ข1 and ๐ข2 are two additional control variables corresponding to the control
accelerations of the spacecraft in the direction of ๐ฅ and ๐ฆ, respectively. The resulting control matrix ๐ต is:
๐ต = [
0 00 01 00 1
] โ โ4ร2 (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
๐ข = 0 therein, ๏ฟฝฬ๏ฟฝ = ๐(๐ฅ, ๐ข) = 0. Three of the points are collinear with the ๐ฅ-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 ๐ฟ1(๐ฅ, ๐ฆ) which is located between Earth and Moon at ๐ฟ1(0.8369, 0).
Since ๐ฅ3 = ๏ฟฝฬ๏ฟฝ1 and ๐ฅ4 = ๏ฟฝฬ๏ฟฝ2, let ๐ฅ1 and ๐ฅ2 be the outputs of the system (30). The resulting output matrix ๐ถ
is:
๐ฆ = [๐ฅ1
๐ฅ2] = ๐ถ๐ฅ โ ๐ถ = [
1 0 0 00 1 0 0
] โ โ2ร4 (32)
The augmented system as required in (11), ๏ฟฝฬ๏ฟฝ๐ = ๐ด๐๐ฅ๐ + ๐ต๐๐ ๐๐ก๐ข0(๐ข๐) + ๐๐(๐ฅ๐), comes in the form (33)
and if one computes ๐ด at point ๐ฅโ = [๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4]๐ = [0.8369, 0, 0, 0]๐, it can be perfectly controlled
through the technique proposed in section (3), i.e., with control law (15), because the pair (๐ด๐ , ๐ต๐) is
stabilizable, ๐๐๐๐(๐๐) = 6 = ๐๐, and ๐ด๐ has all its eigenvalues in the closed left half plane.
๏ฟฝฬ๏ฟฝ๐ = [๏ฟฝฬ๏ฟฝ๏ฟฝฬ๏ฟฝ๐
] = [๐ด ๐ต02ร4 โ๐ฟ
] [๐ฅ๐ข๐
] + [04ร2
๐ฟ] ๐ ๐๐ก๐ข0
(๐ข๐) + [๐ต๐ข๐ + ๐(๐ฅ)
02ร1] (33)
๏ฟฝฬ๏ฟฝ = [๏ฟฝฬ๏ฟฝ๐
๏ฟฝฬ๏ฟฝ๐] = [
06ร6 06ร2
02ร6 02ร2] [
๐๐
๐ฅ๐] + [
[( ๐๐/๐๐ฅ. ๏ฟฝฬ๏ฟฝ โ ๐ด๏ฟฝฬ๏ฟฝ)๐ , 01ร2 ]๐
02ร1] (34)
Being ๐ฟ1 the final destination of the spacecraft, the reference outputs to be tracked are constant and equal
to ๐ฆ๐(๐ก) = [๐ฅ1, ๐ฅ2]๐ = [0.8369, 0]๐ . Considering a bi-dimensional reference system given by ๏ฟฝฬ๏ฟฝ๐ =
๐ด๐๐ฅ๐ + ๐๐(๐ฅ๐) as required in (6), the reference outputs ๐ฆ๐ = ๐ถ๐๐ฅ๐ are easily obtained putting ๐ด๐ = 02ร2
and ๐๐(๐ฅ๐) = 02ร1 , ๐ถ๐ = ๐ผ2 , and initializing it with ๐ฅ๐,0 = [๐ฅ1, ๐ฅ2]๐ = [0.8369, 0]๐ . This way once
๏ฟฝฬ๏ฟฝ๐ = 0 one has ๐ฆ๐(๐ก) = ๐ฅ๐(๐ก) = ๐ฅ๐,0.
Simulation:
In what follows the augmented systems (33) and (34) are solved simultaneously through the RK-Butcher
method between ๐ก0 = 0 and ๐ก๐ = 180 tu, with a step of โ๐ก = 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 ๐ก๐ข๐๐= 150 tu to redirect the spacecraft to the Lagrange point ๐ฟ1. 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, ๐ก should be multiplied by ๐กโ, 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: ๐๐ธ 5.972 ร 1024 kg
Moon mass: ๐๐ 7.346 ร 1022 kg
Mass parameter: ๐ 0.01215
Average Earth-Moon distance: ๐โ 384400 km
Characteristic time: ๐กโ 3.752 ร 105 s , 4.342 days
Gravitational constant: ๐บ 6.674 ร 10โ11 Nm2kgโ2
Table 1. Parameters of the Earth-Moon system.
Initial conditions:
๐ฅ0 = [๐ฅ1, ๐ฅ2, ๐ฅ3, ๐ฅ4]๐ = [0.8369, 0, โ0.0976, 0]๐ , ๐ฅ๐,0 = [๐ฅ1, ๐ฅ2]
๐ = [0.8369, 0]๐
๐ข๐ ,0 = โ๐ข๐ , ๐๐,0 = [๐ต๐ข๐ + ๐(๐ฅ0)
02ร1]
(35)
Controller parameters:
ํ = 1.0 , ๐พ = 1.5 , ๐ = ๐ผ2 , ๐ = ํ. ๐ผ6 (36)
Actuator constraints:
โ0.3 โค ๐ข1 โค 1.8
โ2.0 โค ๐ข2 โค 0.2 ,
|๏ฟฝฬ๏ฟฝ1| โค 180
|๏ฟฝฬ๏ฟฝ2| โค 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 ๐ฟ1 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 ๐ก โฅ ๐ก๐ข๐๐= 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]
Posiรงรฃo: 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]
Posiรงรฃo: (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]
Aceleraรงรตes 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 Aceleraรงรตes 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, ํ, 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, ๐พ, 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 ํ and ๐พ, 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 ๐ ,
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 ๐ฟ1 ,
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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