Robust Control of Chaotic Systems with
Magnitude and Rate Actuator
Constraints
Kouamana Bousson
University of Beira Interior, Portugal
Carlos Velosa
University of Beira Interior, Portugal
ABSTRACT
The present chapter proposes a robust control approach for the class of chaotic systems subject to
magnitude and rate actuator constraints. The approach consists in decomposing the chaotic system into a
linear part plus a nonlinear part and form an augmented system comprising the system itself and the
integral of the output error. The resulting system is posteriorly seen as a linear system plus a bounded
disturbance and two robust controllers are applied: firstly, a controller based on a generalization of the
Lyapunov function, and secondly, the LQR with a prescribed degree of stability. Numerical simulations
are performed to validate the approach applying it to the Lorenz system and to an aeroelastic system, and
parameter uncertainties are also considered to prove its robustness. The results confirm the effectiveness
of the approach and the constraints are guaranteed as opposed to other control techniques which do not
consider any kind of constraints.
1. INTRODUCTION
The problem of control in the presence of magnitude and rate actuator constraints has received a large
amount of attention over the past years but still very few control techniques are opened to community and
the few that exist are kept secret by top companies such NASA, ESA, Airbus, Boeing, among others. On
the other hand, one has noticed that the class of chaotic systems in particular has been controlled
successfully by several control techniques but actuator constraints are almost never or very rarely taken
into account. It is well-known, and it can also be proved whether through analytical, numerical or
experimental techniques, that chaotic oscillations may arise in a wide range of engineering problems such
as in aerospace systems, aeronautical systems, electronic circuits, optical systems, among others, if the
system concerned is subject to specific disturbances. There is therefore a need to formulate control
strategies to control chaotic systems considering also actuator constraints, and, moreover, those strategies
must be robust against parameter uncertainties and disturbances given the particular characteristics that
chaotic systems have.
The objective of the chapter is to propose a robust, easy to implement, and low computational
methodology for the output control of chaotic systems assuming bounded controls. Specifically, it is
proposed a control subject to magnitude and rate actuators constraints - the type of constraints that most
of control engineering applications require, as opposed to other control techniques for this class of
systems which do not consider any kind of constraints. Given the importance of the topic for today’s most
advanced engineering problems, the chapter includes practical examples of the application of the method
to real-world physical systems. This way, one believes that the approach of control proposed is an
interesting tool for the target audience: practicing control engineers, graduate, postgraduate students and
researchers.
The chapter is organized as follows: the problem to be solved is stated in section 3; in section 4, a solution
to the problem, which is a robust control with actuator constraints based on a generalization of the
Lyapunov function and on the LQR with a prescribed degree of stability, is proposed; in section 5,
numerical simulations are presented to validate the approach - it is used to control the classic Lorenz
system and an aeroelastic system, and in both applications parameter uncertainties are also considered to
prove the robustness of the control; section 6 addresses possible future research directions; and, lastly, the
chapter ends with the conclusion of the approach proposed in section 4.
2. BACKGROUND
Chaotic systems are deterministic nonlinear systems mostly with a relatively simple structure, which
despite being governed by well-defined dynamical laws, exhibit an unpredictable motion and are
characterized as being highly sensitive to the initial conditions and to parameter changes. The same
chaotic system started at nearly initial states or with slightly different parameters produces trajectories in
the phase-space which quickly become uncorrelated. Due to this, it is practically impossible to construct
two identical chaotic systems in laboratory without an active control because the trajectories will never be
equal or even similar.
Systems with a chaotic behaviour can arise in a large variety of fields such as in engineering problems,
economics, biology, chemistry, medicine, among others (Boccaletti, Grebogi, Lai, Mancini, & Maza,
2000; F. Chen, Zhou, & Chen, 2011; Ferreira, de Paula, & Savi, 2011; Moon, 1992; Schuster, 1999), and
it can also be proven, through analytical methods, numerical techniques or experimental tests, that any
nonlinear system may exhibit chaotic motions even if the parameters have been previously chosen to
trigger a regular behaviour. It sounds unthinkable at the first glance but that can really be possible if the
system is exposed to external disturbances with particular characteristics. A small perturbation with a
specific power and frequency(ies) is sufficient to trigger a chaotic dynamics. As an example, the attitude
of a satellite in a circular orbit near the equatorial plane of the Earth, despite being planned to exhibit a
regular motion as it happens with the communications satellites, it may become chaotic if the spacecraft
has magnetic elements. The terrestrial magnetic field acts as perturbing torques that are sufficient to
trigger a chaotic attitude (Cheng & Liu, 1999). The fact that an undesirable behaviour may appear in
dynamical systems and that can lead to a catastrophe in some circumstances, led the chaos theory and its
control to become an important research field in the last two decades and an increasingly important
subject in the scientific community.
On the other hand, any control system has to deal inevitably with control constraints regardless if the
controlled system exhibits a regular or a chaotic behaviour. The voltages or currents that can be applied to
an electrical circuit are bounded, the range of the control surfaces of an aircraft is bounded, the thrust of a
rocket is bounded, and so on. These are few examples of actuator constraints that should be considered
when designing a controller. Moreover, there are applications where not only the magnitude of the
actuators plays an important role but also their rates: the slew-rate of an operational amplifier is bounded,
the responsiveness of a pneumatic is bounded, valves and pumps have maximum throughputs, etc., which
should also be considered. Otherwise, actuator saturations can lead to a degradation of the performance of
the system or even to a catastrophe scenario in the extreme case. It was what happened in 1992 regarding
to flight control systems. An F-22 Raptor prototype crash occurred due pilot-induced oscillations partly
caused by rate saturation of the control surfaces which in turn induced time-delay effects in the control
loop. In 1993, a similar aircraft crashed occurred with the Gripen JAS 39 where saturations played again a
critical role (Murray, 1999). It was reported that actuator saturations were also blamed as one of several
unfortunate mishaps which led to the 1986 Chernobyl nuclear power plant disaster, where rate limitations
aggravated an already hazardous situation (Stein, 2003).
One of the most common techniques used to control nonlinear systems is the feedback linearization
wherein the objective consists, in a first stage, in formulating a control law such that the closed loop
system becomes linear and then resorting to powerful and well-known techniques of the linear control to
finish the job. Feedback linearization has been also used to control chaotic systems (Fradkov & Evans,
2005; Liqun & Yanzhu, 1998; Shi & Zhu, 2007). However, this technique requires the exact
mathematical model of the system and consequently does not assure robustness of the control against
model uncertainties neither to external disturbances. Since in most real-world physical systems the model
contains parameters that are known only with some degree of accuracy (ex.: geometric parameters, inertia
parameters, friction parameters, among others), and particularly for the class of chaotic systems that are
extremely sensitive to parameter changes, feedback linearization often turns out to be not successful.
Another remarkable technique to control nonlinear systems is the backstepping control. Based on a
Lyapunov function, it resorts to the computation of fictitious controls in intermediate steps which allows
in a last step the computation of the real control law. Oppositely to the feedback linearization where
nonlinearities are completely eliminated, backstepping control is more flexible in the sense that global
stability can be achieved avoiding that “useful nonlinearities” are eliminated. Backstepping control was
also used to control chaotic systems (Idowu, Vincent, & Njah, 2008; Kareem, Vincent, Laoye, & Akinola,
2008; Yassen, 2006), but it has been reported that besides being cumbersome to find the Lyapunov
function, it is also sensitive to parameter changes. A successful control can be achieved but the model has
to be free of noise.
When a controller is designed based purely on the mathematical model of the system such is the case of
the feedback linearization or the backstepping control, the control law is formulated based on a nominal
model of the physical system. As a result, if the controlled system contains uncertainties on its dynamics
or if it is subject to external disturbances, the controller stops being effective precisely because it is
designed assuming that the model is accurate and does not suffer any change. Taking into account that
uncertainties are almost always present in most dynamical systems (ex.: uncertainties in parameters,
noises in sensors, unmodeled dynamics, undesired fluctuations in actuators), the robust control, whose
control law is designed considering not only the nominal model but also some uncertainties, is a very
effective tool. SMC (Sliding Mode Control), LQG/LTR (Linear-Quadratic-Gaussian / Loop Transfer
Recovery), QFT (Quantitative Feedback Theory), -Synthesis Control and
are techniques which
can be used to ensure a robust control. Robust control has been also used to control chaotic systems
(Huang & Feng, 2008; Xu & Yang, 2009; Yan, Yang, Chiang, & Chen, 2007). Nevertheless, robust
control has limitations when used to control systems with actuator constraints.
Adaptive control is an excellent technique to deal with systems subject to uncertainties or whose
parameters are time-varying. Oppositely to robust control that guarantees that the control law needs not to
be modified if the uncertainties are within pre-specified limits, adaptive control does not require any a
priori information about parameter or uncertainty limits, and the control law is constantly modified over
the time. MRAC (Model Reference Adaptive Control), STR (Self-Tuning Regulator), Gain Scheduling
and Dual Control are examples of adaptive techniques that can be used to control uncertain models.
However, dealing with uncertainties is not an easy task and in that sense adaptive control has a drawback:
the structure of the model parameters must be known. Such requirement may limit its application to the
control of chaotic systems. For example, it was not yet shown that two oscillators with very different
structures can be synchronized through adaptive control (Femat & Solis-Perales, 2008). Even so, chaotic
systems have been also controlled through this technique (Alstrom, Marzocca, Bollt, & Ahmadi, 2010;
Ayati & K-Sedigh, 2008; Koofigar, Sheikholeslam, & Hosseinnia, 2011) but, once again, not considering
actuator constraints.
MPC (Model Predictive Control), also known as RHC (Receding Horizon Control), is one of the most
advanced and successfully techniques for the control of dynamical systems. Its principle consists in the
successive resolution in real-time of a finite horizon optimal control problem based on measurements or
estimates of the current system states. The major advantage of predictive control lies in the possibility to
include several restrictions on the optimization criterion such as output constraints, state constraints, and
mostly important for the present work, control magnitude constraints and constraints on the variation of
those signals. At first glance, MPC seems to be the best technique to achieve the purpose proposed in this
chapter. Nonetheless, although it has been used also to control chaotic systems (Ahn, Lee, & Song, 2012;
Mohammadbagheri & Yaghoobi, 2011; Polyak, 2005) the models concerned are free of noise and the
control may fail. On the other hand predictive control may require, particularly for systems with a fast
dynamics, a large power of computational calculus because the resolution time of the optimization
problem between each sample becomes a requisite that can be on the order of milliseconds or even
microseconds.
Techniques specifically designed to control chaotic systems such as the renowned OGY method (Ott,
Grebogi, & Yorke, 1990), the Continuous Delayed Feedback Control (Pyragas, 1992) and the Otani-Jones
Control (Otani & Jones, 1997) have been used successfully in practise although having some
unfavourable points. One of the drawbacks that stands out most is that actuator constraints are not taken
into account.
Control based on LMIs (Linear Matrix Inequalities) is an interesting control approach and it has been also
used to control chaotic systems (Deng & Xu, 2010; Tanaka, Ikeda, & Wang, 1998; Yu, Zhong, Li, Yu, &
Liao, 2007). The technique allows calculating the gain of the system in such a way that the controlled
system satisfies certain performance requirements. That includes the possibility to consider stability
conditions, conditions for a specific decay rate, conditions in order to reduce the effect of disturbances on
the outputs (disturbance rejection), output constraints, and also input constraints. It is an effective
approach but at same time it requires a special care and also some experience by the designer because
some parameters should be relaxed in order to obtain an admissible control. When using toolboxes to
solve LMIs, the decision variables are optimized by the resolution algorithms, and sometimes it is
necessary to relax some parameters so that the system responds as intended. If used to control a chaotic
system subject to control constraints it may happen that saturation occurs so easily, when actually it is
needless, and can lead to instability.
A common approach to deal with actuator constraints consists in designing a controller considering no
input constraints and in adding a saturation block after the controller output followed by anti-windup
techniques (Hippe, 2006). However, this is not the best approach because it introduces nonlinearities on
the system, degrades its performance and in some cases can lead to instability.
To sum up, several control techniques have been used to control chaotic systems. However, this class of
dynamical systems in particular requires a special attention due to its characteristics. Essentially, due to
its high sensitivity to initial conditions and to its high sensitivity to parameter changes. Its control should
be therefore robust against parameter uncertainties in order to be effective. On the other hand, one has
noticed that there is no much work opened to community regarding to control of chaotic systems with
actuator constraints. Thus, this chapter contributes in that sense. Not only with robust control with
magnitude constraints but also with rate constraints.
3. PROBLEM STATEMENT
Consider an output tracking problem of a time-invariant nonlinear system in a continuous-time approach.
Let that system be described by differential equations of the form (1) and the reference signals to be
tracked generated by the reference system (2):
(
̇
Controlled system:
̇
Reference system:
)
(
(1)
)
(2)
where
,
represent state vectors,
,
vectors of parameters,
the
outputs of each system (vectors with same dimension) where are the reference signals to be tracked,
the control input vector,
,
the output matrices,
the input matrix,
two smooth nonlinear functions, and ̇
.
For the class of chaotic systems, a nonlinear system of the form (1) can exhibit a regular motion (ex.:
equilibrium points, periodic orbits, multi-periodic orbits) or a chaotic motion depending on the values of
the parameters. Lorenz, Chua, Rössler, Chen, Genesio-Tesi and Coullet systems are well-known
examples of that. On the other hand, even if the parameters have been set to trigger a regular behaviour, it
can be proven through a Melnikov analysis that the trajectory of any nonlinear system may become
chaotic if it is subject to external disturbances with particular characteristics. Chaos may be desirable or
not depending on the application, but an unexpected behavioural change is no longer desirable.
Most of control approaches to control systems of the form (1) such that their outputs follow given
reference signals consider that the parameters are kept constant. However, in practical applications, the
parameters can suffer unavoidably slight variations due to external causes which in turn can force the
system to exhibit a different behaviour from the desired one. As an example, consider the well-known
Chua circuit whose parameters are set by electronic components (resistors, inductors and capacitors). If
the parameters are chosen so that the system exhibits a chaotic oscillation, for example to be applied in a
secure communication, it is possible that the oscillation can be no longer chaotic because chaotic systems
are extremely sensitive to parameter changes, and factors such as temperature, humidity and age cause
variations in the component values.
On the other hand, real-world physical systems have to deal necessarily with actuator constraints. The
classic approach to limit the magnitude of the control consists in adding a saturation block at the
controller output and resort to anti-windup techniques to prevent both controller and plant windups.
Nevertheless, despite being a workable solution, the approach is not the best because it introduces
nonlinearities in the system and the resulting performance is not the best. Moreover, due to
electrical/mechanical restrictions, some applications require not only magnitude constraints but also rate
constraints.
The problem to be solved consists in forcing the outputs of the system (1) to follow bounded reference
signals even if the parameters are not known accurately and considering both magnitude and rate actuator
constraints. Mathematically, the purpose is to find ( ) such that conditions (3) and (4) remain:
Problem to solve: Assume ‖ ( )‖
and find ( ) such that:
‖ ( )‖
for any
wherein
, and
‖ ( )
( )‖
(3)
‖ ( )‖
, ‖ ̇ ( )‖
from the time-instant at which the control is turned on
(4)
.
In inequality (3), ‖ ‖ represents an appropriate norm and
is desirable to be as small as possible for
a good tracking. Parameters
and
denote respectively the magnitude and rate saturation
bounds of the control vector.
4. THE APPROACH PROPOSED
Consider a dynamic system of the form:
̇
(
)
(
) ,
( )
(5)
where ( )
for all , is continuously differentiable in and in the components of for near the
origin, and is a bounded disturbance continuously differentiable with respect to the components of
and Lebesgue measurable with respect to .
Definition 1: Stability in the Sense of Lyapunov
The unperturbed system (5), that is, with
therein, is said to be stable in the sense of Lyapunov with
respect to the equilibrium
, if for any
and for any initial time
, there exists a positive
(
)
constant
, such that:
‖ ( )‖
‖ ( )‖
, for all
(6)
Definition 2: Asymptotic Stability
The unperturbed system (5), that is, with
therein, is said to be asymptotically stable about its
equilibrium
, if it is stable in the sense of Lyapunov and, furthermore, there exists a positive
(
)
constant
, such that:
‖ ( )‖
( )
(7)
Definition 3: Uniform Stability
The unperturbed system (5), that is, with
therein, is said to be uniformly stable about its
equilibrium
, if it is stable (asymptotically or in the Lyapunov sense) with the constraint being
( ).
independent of ,
Definition 4: Stability under Persistent Disturbances
The perturbed system (5) is said to be stable under persistent disturbances (or also totally stable) about
its equilibrium
, if given a tolerance
there are two positive constants, and , such that:
‖ ̃( )‖
‖ ( ̃ )‖
}
‖ ̃( )‖
where ̃( ) denotes the solution of the perturbed system.
Theorem 1 (Malkin Theorem):
, for all
(8)
If the unperturbed system (5) is uniformly and asymptotically stable about its equilibrium
, then it
is stable under persistent disturbances. Namely, the persistently perturbed system remains to be stable in
the sense of Lyapunov, (G. Chen, 2004; Hoppensteadt, 2000).
Chaotic systems have by nature a simple structure. That is, the differential equations are constituted
mainly by linear terms and by some nonlinear terms. In that sense, the controlled system (1) can be
decomposed into a linear part plus a nonlinear part as follows:
( )
̇
(9)
where, in turn,
can be seen as a state matrix and ( ) as disturbances. The parameter vector
was dropped to simplify the terminology but the parameters still implicitly on matrix and/or on .
Remark:
For nonlinear systems wherein the linear part
does not exist explicitly, (
) in (1) can be written
into the same form of (9) linearizing it about a specified state and writing the remaining terms as:
( )
( )
. Typically, the origin
can be chosen as the linearization point. However, some
precautions must be taken into account. There are some particular nonlinear systems which are
controllable at the origin but that when linearized about the same point are no longer controllable. It is the
case of the underactuated systems. In these specific systems, the point should be chosen in such a way
that the linearized system still to be controllable. Alternatively, an optimal linearization could be
performed instead of the classic linearization (Jacobian matrix). With the optimal linearization the
controllability characteristic is not lost during the linearization process, (Bousson & Quintiãs, 2008).
Define a vector
as the integral of the output error:
∫ (
)
(10)
Let
denotes an operator which will be used to enforce the effective control input
( ) to satisfy
amplitude and rate constraints of the form (4). Let that operator be given by equation (11), (Saberi,
Stoorvogel, & Sannuti, 2011):
( )
(11)
where:
̇
( )
and
( )
( (
))
(12)
( ( ))
is a saturation function of limit , defined as:
( )
{| | }
( )
(13)
Consider now an augmented system composed by (9) and by the time-derivative of equation (10).
Together with the operator (11-13), that system yields:
̇
[ ]
̇
where can be written in the form:
[
][ ]
[
]
( )
[
( )
]
( )
(14)
̇
with
(
)
denoting the augmented state vector,
the augmented input matrix and
( )
(
)
(
) (
(15)
)
the augmented state matrix,
the vector of external disturbances.
At this point, the output tracking problem with constraints (3-4) was transformed into a regulation
problem of the form (14), where the goal is to find the control ( ) such that ( ̇ ̇ ) ( ). Note that
this implies, in turn, that ( )
( ).
One of the characteristics of chaotic systems is that their trajectories in the phase space are bounded. If
one considers that the reference signals to be tracked are also bounded, all components of the disturbance
vector (
) remain consequently bounded. Assuming that is a continuous-time vector function, and
actually it is, one has, by definition (continuous and bounded function), that is Lebesgue measurable
with respect to . Therefore, a robust controller can be design to stabilize asymptotically the unperturbed
system (15), that is, with
therein, because the persistently perturbed system will remains stable in
the sense of Lyapunov, (theorem 1).
Note that if a time-invariant system, which is the case of the unperturbed system (15), is designed to be
asymptotically stable, it will be asymptotically- and also uniformly stable because a time-invariant system
has the constraint independent of .
The robust control approach proposed in this chapter to stabilize the system (15) is presented in the
following subsections and is divided in two parts: a control based on the Korobov technique (Korobov &
Skoryk, 2002) and a control through an extension of the LQR - the LQR with a prescribed degree of
stability (Naidu, 2002). In both cases, it is assumed that the augmented system (15) is controllable, or in
other words, that the pair (
) is controllable which implies that the rank of the controllability matrix
is equal to the order of the system (15):
[
]
(16)
4.1. Korobov control
(Korobov & Skoryk, 2002) developed a control law based on a controllability function (a generalization
of the Lyapunov function) that steers, under certain conditions, the trajectory of linear systems of the form
̇
to the origin. The technique takes into account control constraints which can be chosen
until a given order, that is, constraints in
̇ ̈ , but the trajectory must to start inside a small
neighbourhood of the origin so that those constraints are satisfied. Nevertheless, the assumptions made by
Korobov allow realizing that the controlled trajectory is steered to the origin even if its initial point has
not started on that attraction domain and even if the model is perturbed. Bounded and Lebesgue
measurable disturbances are acceptable. The drawback that arises as a consequence is that the saturation
limits of the controls are no longer satisfied.
Once the trajectory can be far away from the reference when the control is turned on, alone, this technique
( ),
has not power to ensure a constrained control. Therefore, one attached the operator (11-13),
which is already included in the augmented system (15), to guarantee an effective constrained control.
Consider the augmented system (15). Let the fictitious control
(Korobov & Skoryk, 2002):
be computed by the control law (17),
(
where the way to find matrices
,
,
( ) and
( )
)
(17)
is resumed below:
is an upper triangular matrix of dimension
and the remaining defined as
the vector is the th column of the matrix
condition (18):
in which the elements of the main diagonal are
,
. For the calculation of each element
,
, the values of are chosen so as to satisfy the
∑
(
)
(18)
and each vector ,
, is chosen so as to be orthogonal to all linearly independent vectors (19)
⟩
with the exception of the vector
, and such that the scalar product ⟨
is satisfied.
(19)
is a matrix defined by (20):
(
is a matrix of dimension
each is defined for
as
)
(
with elements (
∑
.
)
let
be a matrix of dimension
given by
is a matrix in which the elements of the first off-diagonal equal
)
(20)
and the others equal to zero, wherein
(
) where each
and all the other elements equal .
( ): is given by the inverse matrix of (21) where the parameter can assume any value
. Here, denotes the order of the derivative of the control and one considers
for constraints
on the magnitude and on the rates.
( )
Controllability function (
the equation (22):
∫
(
)
): the controllability function
⟨
( )
where ⟨ ⟩ denotes the dot product, and its unique solution
following conditions:

( )

(
.
)
for all
(21)
.
(
⟩
) is defined, for any
, by
(22)
is a scalar function that satisfies the
(
 ∑

(
)
(
(
) is continuous at all
In equation (22),
( (
))
)) ,
and continuously differentiable at any
is a scalar number chosen within the interval:
̅
where matrix
and
:
except at the origin.
‖
is computed by the integral (24),
(23)
‖
defined by the expressions (25-27),
∫
‖
,
(24)
‖ (∑
‖ ‖
‖)
‖
(25)
(26)
(
‖
)
‖
∏
In (25),
{
} and
if
matrix of the dimension of and in (27)
(
(27)
)
and
if
( ) calculated with
. In (26),
.
denotes the identity
In practise, the solution that satisfies relation (22) is obtained numerically and the calculation process
throughout the entire trajectory
may become time consuming. This can be seen as a drawback because
the frequency at which the control can be applied to the system is limited by the time necessary to
compute . However, it can be proven (Korobov & Skoryk, 2002) that the controllability function
( ) is equal to the solution of the differential equation ̇
if its initial condition is equal
⟨
( )
⟩
to
. Thus, expression (22) can be solved only once,
and once
̇
obtained the solution
( ( )),
can be solved simultaneously with system (15) and ( )
substituted by ( ( )) when calculating the control.
The control law (17) has the power to steer the trajectory of the system (15) to the equilibrium state but
has not the power to hold it after that. That is, if the equilibrium is the origin, the trajectory is steered just
until reach the point immediately before. This is why there is a need to switch to another controller when
the equilibrium is reached. As a criterion to do that, let us define a tolerance
and carry out the
switch when the value of the controllability function is less than or equal to , ( )
. Note that as
approaches to zero means that the controlled trajectory approaches to the reference.
4.2. LQR with a specified degree of stability
For a linear time-invariant system of the form ̇
with initial condition ( )
and
no final time constraint, it is well-known from the optimal control theory that the control law which steers
the trajectory to the origin minimizing the performance index:
(
∫
)
,
(28)
is given by (29), (Naidu, 2002):
(29)
where
(ARE):
is a symmetric and positive definite matrix, solution of the Algebraic Riccati Equation
(
)
(
)
wherein
and
are two weighting matrices, being
definite (
) and symmetric and positive definite (
number denoting the specified degree of stability.
(30)
symmetric and positive semi), and
a scalar
Regarding to the system to be controlled, system (15), let the fictitious control be computed by the
control law (29) from the time-instant at which the trajectory ( ) is close to the equilibrium state.
Under this condition, both trajectories ( ) and ( ) are close to each other, and as a result, the effort of
the control required to hold the output error small is not too large. The largest effort occurs when the error
is large but for that situation the Korobov control undertakes to decrease it.
The robustness of the LQR is assured by the parameter . Once all components of the disturbance vector
(
) are bounded and Lebesgue measurable,
can be chosen such that the controller supports
such disturbance because causes a -shift in the eigenvalues of the closed-loop matrix ̅
to the left in the direction of the real axis.
( ) applied to
Note that from the time-instant at which the LQR is turned on the effective control
system (15) does not saturate, and in that sense, one can say that the control is optimal in the sense it
minimizes the performance index (28).
5. SIMULATION RESULTS
In this section one considers two engineering applications and presents numerical simulations to validate
the approach proposed. The first application deals with the output tracking of a classic chaotic system and
the second one with the flutter or chaotic motion suppression of an aeroelastic system. In both
applications the parameters of the controlled system are also changed over the time in order to validate
the robustness of the control.
5.1. Application 1: Lorenz system
It was considered two well-known chaotic systems, Lorenz and Chua, and the goal is to force the outputs
of the Lorenz system to follow reference signals generated by the outputs of the Chua system. In a first
stage, the parameters of both systems are set to nominal values which lead them both to exhibit chaotic
motions and the methodology is applied to validate its effectiveness when tracking chaotic signals. In a
second stage, the parameters of the Lorenz system are changed over the time with the purpose to verify
the robustness of the control against parameter uncertainties. Here, the parameters of the Chua system are
kept unchanged, once what the controller needs are the reference signals and not the reference model.
The Chua system is a nonlinear chaotic oscillator described, in its normalized form, by the following set
of differential equations (Chua, Wu, Huang, & Zhong, 1993):
̇
(
(
))
̇
(31)
̇
where:
( )
(
)|
|
|
|
The parameters of system (31) are
and for the values presented below the system exhibits the
chaotic behaviour shown in figure (1), a double-scroll attractor:
,
,
,
50
4
45
3
40
2
35
30
x
3
1
3,r
(32)
Lorenz attractor
Chua attractor
x
,
25
0
20
-1
15
-2
0.5
-3
5
0
-4
-3
10
-2
-1
0
x 1,r
1
2
3
x 2,r
-15
-0.5
Figure 1. Phase-space of the Chua system.
Considers that the first state variable
unique reference signal to be tracked:
40
20
0
-20
0
-10
-5
0
x1
5
-20
10
15
20
x2
-40
Figure 2. Phase-space of the Lorenz system.
( ) is the unique output
( ) of system (31) and therefore the
(33)
The Lorenz system is a simplified mathematical model of atmospheric convection governed by the
following set of differential equations (Lorenz, 1963):
(
̇
)
̇
(34)
̇
which for the values of parameters
(2):
given below gives rise to the chaotic motion shown in figure
,
,
Decomposing the Lorenz system into a linear part plus a nonlinear part, yields:
(35)
̇
[ ̇ ]
̇
[
][ ]
[
]
(36)
where, if one considers two control inputs,
on the first state equation and
trajectory of the Lorenz system becomes controllable and the system written as:
̇
on the third one, the
( )
(37)
with:
[
]
[
,
]
(38)
( ) is the unique output ( ) of system (36):
Considers that the first state variable
(39)
Writing the Lorenz system together with the integral of the output error and applying the operator
to ensure bounded controls, one has the augmented system:
̇
[ ]
̇
[
][ ]
[
]
( )
[
( )
]
( )
( )
(40)
which takes the form (15) and is ready to be controlled with control laws (17) and (29).
Note that both the Lorenz as the Chua systems are both chaotic and therefore the term (
) is a
bounded disturbance and Lebesgue measurable with respect to as assumed in (5). On the other hand,
system (40) can be controlled because the pair (
) is controllable, that is, the rank of the
controllability matrix is equal to its order,
( )
.
For simulation purposes, the ordinary differential equations of systems (12), (31) and (40) were solve
together using the RK-Butcher method between
and
seconds, with a step of
s,
and with initial conditions (41). In (12), the value of the required parameter is
. The parameters
of the Korobov controller and of the LQR are the ones presented in (42) and switching from one to
another occurs when the controllability function is ( )
. The constraints of the actuators are
presented in (43). To ensure that both systems, Lorenz and Chua, are already in a chaotic regime the
control was turned on at
s, a time-instant wherein the transient period has already gone.
Initial conditions:
,
,
,
(41)
Controller parameters:
Korobov
LQR
,
(42)
Actuator constraints:
‖ ‖
,
‖ ̇‖
(43)
Simulation 1:
In the following simulation the parameters of both systems are kept unchanged. Figure (3) shows the time
evolution of the state variables and of the outputs of each system. The first vertical dashed line indicates
the time-instant at which the Korobov control is turned on and the second one the time-instant at which
the control is switched to the LQR. The output error
is shown in figure (4). The magnitude
and rates of the control variables are shown in figures (5) and (6), respectively. The rates / derivatives of
the control variables were computed numerically by the method of the centered finite differences. Figure
(7) shows the integral of the output error , and figure (8) the controllability function of the Korobov
control.
x 1 = y Lorenz
x 1 = y Chua
ey
40
30
20
0
20
-20
-40
0
10
x 2,Lorenz
20
x 2,Chua
30
t [s]
40
50
60
10
40
20
0
0
-20
-40
0
10
x 3,Lorenz
20
x 3,Chua
30
t [s]
40
50
60
-10
60
40
-20
20
0
-20
0
10
20
30
t [s]
40
50
60
-30
0
10
Figure 3. State variables: Lorenz and Chua.
20
30
t [s]
40
50
Figure 4. Output error:
300
60
.
3000
u1
u1,dot
200
2000
100
1000
0
0
-100
-1000
-200
-300
0
10
20
30
t [s]
40
50
60
140
-2000
0
10
20
30
t [s]
40
50
60
20
30
t [s]
40
50
60
2000
u2,dot
u2
120
1500
100
1000
80
60
500
40
0
20
-500
0
-20
0
10
20
30
t [s]
40
Figure 5. Control variables: ‖ ‖
50
60
.
-1000
0
10
Figure 6. Control rates: ‖ ̇ ‖
.
45
100
(t)
q
40
50
35
30
0
25
20
-50
15
-100
10
5
-150
0
-200
0
10
20
30
t [s]
40
50
60
Figure 7. Integral of the output error.
-5
0
10
20
30
t [s]
40
50
60
Figure 8. Korobov controllability function.
Simulation 2:
In this simulation the parameters of the Lorenz system are slightly changed over the time to demonstrate
the robustness of the control against parameter uncertainties. One considers that the parameters are known
within specified intervals, which is what happens in most real applications. Considering that each
parameter may suffer a change around its nominal value in about
, one has
,
and
. Let the parameter vector
be perturbed according with a
multivariate normal distribution with their nominal values as means and covariance matrix computed such
that each parameter belongs to its respective interval with
sure. (
, where denotes the
mean of each parameter and the respective standard deviation). The correlation coefficients assumed
between each two parameters are the ones presented in (45):
Mean vector of parameters and vector of standard deviations:
[ ]
[
]
[ ]
,
[
]
(44)
Pearson correlation coefficients:
,
,
(45)
Covariance matrix:
[
]
[
]
(46)
Figure (9) shows the time evolution of the perturbed parameters. Figure (10) shows the state variables of
the Lorenz and of the Chua systems, figure (11) the magnitude of the control variables and figure (12)
their respective rates.
 Lorenz
x 1 = y Lorenz
12
40
11
20
10
0
9
-20
8
0
10
20
Lorenz
30
t [s]
40
50
60
-40
40
30
20
28
0
26
-20
0
10
20
Lorenz
30
t [s]
40
50
60
-40
60
2.8
40
2.6
20
2.4
0
0
10
0
10
x 3,Lorenz
3
2.2
10
x 2,Lorenz
32
24
0
20
30
t [s]
40
50
60
-20
0
Figure 9. Perturbed parameters (Lorenz).
10
x 1 = y Chua
20
x 2,Chua
20
x 3,Chua
20
30
t [s]
40
50
60
30
t [s]
40
50
60
30
t [s]
40
50
60
Figure 10. State variables: Lorenz and Chua.
300
3000
u1
u1,dot
200
2000
100
1000
0
0
-100
-1000
-200
-300
0
10
20
30
t [s]
40
50
60
140
-2000
0
10
20
30
t [s]
40
50
60
20
30
t [s]
40
50
60
2000
u2
120
u2,dot
1500
100
1000
80
60
500
40
0
20
-500
0
-20
0
10
20
30
t [s]
40
Figure 11. Control variables: ‖ ‖
50
60
.
-1000
0
10
Figure 12. Control rates: ‖ ̇ ‖
.
5.2. Application 2: Aeroelastic system
In the aeronautical field an aeroelastic system refers to an aircraft system including a wing or a tail in
which the interaction of the elastic structure, the inertia and the aerodynamic forces can cause many
complicated phenomena presumably due to nonlinear effects of the structure. It is well-known that from a
specific/critical flight speed, a limit cycle oscillation, which typically is referred as flutter, occurs. The
phenomenon arises either in subsonic flow as in supersonic flow, and under some particular conditions
can degenerate to chaos, (Alstrom et al., 2010; Bousson, 2010; F. Chen et al., 2011; Ramesh &
Narayanan, 2001; Wang, Chen, & Yau, 2013; Zhao & Yang, 1990; Zheng & Yang, 2008). Furthermore,
if the controlled aeroelastic system is subject to actuator constraints, structural instabilities may also arise
(Demenkov, 2008). Because limit cycle oscillations and possible chaotic motions are dangerous
phenomena which can lead eventually to a structural failure due to material fatigue and consequently to
an aircraft disaster in the worst scenario, a robust controller is needed to supress any undesired motion
during the flight. Moreover, the controller must deal with actuator constraints because it is well-known
that the amplitudes and the velocities at which the control surfaces can operate are bounded due to
mechanical, electrical or aerodynamic limitations.
In this sub-section, one applies the control approach proposed in section (4) to suppress effectively the
undesired motions of an aeroelastic system. Following references (Platanitis & Strganac, 2004; Wang et
al., 2013), the model of a two-degree of freedom wing structure includes a linear spring oriented along the
plunge displacement, a rotational spring along the pitch angle, corresponding dampers, and at a given
flight speed the wing oscillates along the plunge displacement direction and rotates at the pitch angle
about the elastic axis. The control of both motions is done through two control surfaces: one at the leading
edge and one at the trailing edge. Figure (13) shows the respective two-dimensional aeroelastic system.
Figure 13. Aeroelastic system with two degrees of freedom.
The mathematical model for the aeroelastic system depicted in figure (13) is (Wang et al., 2013):
̈
][ ]
̈
[
̇
][ ]
̇
[
[
][ ]
( )
[
]
(47)
where the plunge displacement and pitch angle are the variables to be controlled and and the
angles of the leading- and trailing-edge of the control surfaces deflection, respectively.
The aerodynamic terms and (lift force and moment respectively) can be modelled suitably for a lowfrequency and subsonic flight scenario including a leading-edge and a trailing-edge control surface as
(48), the well-known Theodorsen functions:
[
̇
(
[
(
(
̇
)
̇
)]
(48)
(
)
̇
)]
where is the air density, the flight speed, the wing section half-chord, the wing section span,
the dimensionless distance from the mid-chord to the elastic axis position,
and
the lift and
moment coefficients per unit angle of attack, respectively,
and
the lift and moment coefficients
per unit angle of wing trailing-edge control surface deflection, respectively,
and
the lift and
moment coefficients per unit angle of wing leading-edge control surface deflection, respectively, and
,
,
are the effective control moment derivatives due to angle of attack and
trailing- and leading-edge control surfaces deflection respectively about the elastic axis, defined as:
(
)
(
,
)
(
,
)
(49)
In system (47),
is the total mass of the pitch-plunge system (wing plus suporting body),
the mass
of the wing section,
the dimensionless distance between the center of gravity and the neutral axis,
the moment of inertia (where
denotes the pitch cam moment of
inertia,
the wing section moment of inertia about the center of gravity and
), and
and the plunge displacement and the pitch angle damping coefficients, respectively, and
and ( )
the plunge displacement and the picth spring stiffness coefficients, respectively.
̇,
Defining the state variables:
,
of ordinary differential equations as:
̇ , system (47) can be converted into the form
,
̇
̇
̇
̇
(50)
̇
̇
̇
̇
which in turn can be written into the form of a linear part plus a nonlinear part ̇
( ):
̇
̇
[ ]
̇
̇
with the coefficients
()
̇
̇
̇
̇
[
][ ]
[
]
(51)
]
(
,
(
)
)
)(
{[(
̇
[
defined as:
,
̇
[ ]
(
)
]
}
)
(
)
(
)
(52)
(
)
)(
{[(
̇
̇
(
) ]
}
)
(
)
(
)
Since the purpose of the present application is to supress undesired motions whenever they occur (limit
cycle or chaotic oscillations), the goal in terms of control can be simplified to find the control
that forces the state variables and to be zero instead of forcing the trajectory of system (51) to the
origin. Note that
̇ and
̇ and given the nature of these relations the entire trajectory is
steered to the origin because ( )
.
Being
and
the outputs of system (51), the output matrix
[ ]
yields:
[
]
(53)
and the reference signals to be tracked is this particular case are constant,
augmented system as required in (15), one has:
̇
[ ]
̇
[
][ ]
[
]
( )
[
( )
( )
]
( )
. Forming the
(54)
( )
where it is now in the form of a perturbed system ̇
(
) and can be
perfectly controlled through the control strategy proposed in section (4), i.e., with control laws (17) and
then with (29), because the disturbance term (
) is bounded and the pair (
) is controllable,
( )
.
Hereafter one presents two simulation cases wherein the aeroelastic system is already in a limit cycle
oscillation (the wing section moving vertically and rotating, continuously) and the control is turned on to
supress the undesired motion. For that, the augmented system (54) is solved together with equation (12)
using the RK-Butcher method between
and
s, with a step of
s, and departing
from initial conditions (55). The parameters of the aeroelastic system are the ones stated in table (1) and
the required parameter in (12) is
. The control is turned on at
s, a time-instant at which
the system exhibits already a limit cycle, and switching from the Korobov controller to the LQR
controller occurs when the controllability function is ( )
. The parameters of both
controllers are presented in (56). Regarding to the actuator constraints, one considers that each control
surface, leading-edge and trailing-edge, can has a maximum deflection of
,
, and that
̇
the speed at which each surface can act is
, ̇
.
Parameter
Value
Parameter
Value
(
)
Table 1. Parameters of the aeroelastic system.
Initial conditions:
,
,
(55)
Controller parameters:
Korobov
LQR
,
(56)
Actuator constraints:
‖ ‖
‖ ̇‖
,
(57)
Simulation 1:
In this simulation the free stream velocity is assumed constant and selected such that triggers the flutter
phenomenon/limit cycle oscillation. It was set to
as considered by (Wang et al., 2013).
In figure (14) one shows the time evolution of the state variables of the aeroelastic system. The plots on
the left represent also the system outputs, and . The first vertical dashed line indicates the time-instant
when the Korobov control is turned on and the second one the time-instant when the control is switched
to the LQR. Figures (15) and (16) show respectively the amplitudes and rates of the control surfaces
(leading- and trailing-edge). Figure (17) shows the phase space of the system subdivided in two phase
planes: the plane (
̇ ) on the left and the plane ( ̇ ) on the right.
300
20
200
 dot [º/s]
 [º]
10
0
100
0
-100
-10
-200
-20
0
5
10
15
20
-300
25
0
5
10
t [s]
15
20
25
15
20
25
t [s]
200
15
10
100
[cm/s]
0
dot
0
-5
h
h [cm]
5
-100
-10
-15
0
5
10
15
20
25
-200
0
5
10
t [s]
Figure 14. State variables of the aeroelastic system.
t [s]
10
100
=  [º/s]
dot
200
0
1,dot
u =  [º]
1
20
-20
u
-10
0
5
10
15
20
0
-100
-200
25
0
5
10
10
=  [º/s]
dot
0
2,dot
-5
u
u =  [º]
2
20
25
15
20
25
200
5
-10
-15
15
t [s]
t [s]
0
5
10
15
20
100
0
-100
-200
25
0
5
10
t [s]
t [s]
Figure 15. Control variables: ‖ ‖
Figure 16. Control rates: ‖ ̇ ‖
.
.
200
250
200
150
150
100
100
50
[cm/s]
0
dot
0
h
 dot [º/s]
50
-50
-50
-100
-100
-150
-150
-200
-250
-20
-15
-10
-5
0
5
10
15
-200
-15
20
-10
-5
0
h [cm]
 [º]
Figure 17. Phase space subdivided in two phase planes: the plane (
on the right.
5
10
̇ ) on the left and the plane (
15
̇)
Simulation 2:
To validate the approach against parameter uncertainties one considers in the following simulation a
change over the time in the free stream velocity according with function (58), figure (18). The results
are presented in figures (19) to (21).
(
)
(58)
26
24
V [m/s]
22
20
18
16
14
12
0
5
10
15
20
t [s]
Figure 18. Perturbed free stream velocity.
25
20
300
200
 dot [º/s]
 [º]
10
0
100
0
-100
-10
-200
-20
0
5
10
15
20
-300
25
0
5
10
t [s]
15
20
25
15
20
25
15
20
25
15
20
25
t [s]
15
200
10
100
[cm/s]
0
dot
0
-5
h
h [cm]
5
-100
-10
-15
0
5
10
15
20
-200
25
0
5
10
t [s]
t [s]
200
10
100
=  [º/s]
dot
20
0
1,dot
u =  [º]
1
Figure 19. State variables of the aeroelastic system (with parameter uncertainty).
-20
u
-10
0
5
10
15
20
0
-100
-200
25
0
5
10
t [s]
t [s]
200
25
=  [º/s]
dot
15
2,dot
10
u
u =  [º]
2
20
5
0
0
5
10
15
20
25
0
-100
-200
0
5
10
t [s]
t [s]
Figure 20. Control variables: ‖ ‖
100
.
Figure 21. Control rates: ‖ ̇ ‖
.
6. FUTURE RESEARCH DIRECTIONS
The problem addressed in this chapter concerns the robust control of chaotic systems with bounded
controls. Specifically, the approach proposed deals with the control of this class of systems assuming that
the system to be controlled is subject both to magnitude and rate actuator constraints. Nonetheless,
although the objective has been successfully achieved, there are interesting ideas which are not yet
solved, at least that is authors’ knowledge to date, and that should be improved. Notice that in the
approach proposed the actuators may saturate! However, it is well-known that chaotic systems are highly
sensitive to initial conditions, and in that sense, one believes that is possible to steer the trajectory to a
given point in the phase space through small amounts of control without reach the saturation limits. Thus,
here stays an interesting problem for researchers and engineers: How to control (or synchronize) chaotic
systems without even saturate the actuators?
7. CONCLUSION
In the present chapter a robust control approach is proposed to control the class of chaotic systems
assuming magnitude and rate actuators constraints. The purpose of the control precisely is to force the
outputs of a chaotic system to track specified reference signals. The approach consists in decomposing the
nonlinear system into a linear part plus a nonlinear part and form an augmented system comprising the
system itself and the integral of the output error. Thus, the tracking problem is transformed into a
regulation problem wherein the system to be controlled is seen as a linear system plus a bounded
disturbance. This way, the mathematical model of the reference system (who generates the reference
signals) is not required as opposed to other control techniques. Two robust controllers are posteriorly
designed to stabilize the system about its equilibrium state and an operator is employed to ensure a
constrained control. Two controllers are used because the trajectory of the system can be anywhere in the
phase space when the control is turned on and it is also intended a control with global stability. This is
overcome starting the control with a robust control of Korobov (a control based on a generalization of the
Lyapunov function) and then substituting it by the LQR with a prescribed degree of stability. Numerical
simulations are presented to validate the approach proposed. The control is applied to the classic Lorenz
system and to an aeroelastic system. In both applications some parameters are also slightly changed over
the time to validate the robustness of the control against parameters uncertainties. The results show that
the outputs track successfully the reference signals and that the error remains stable in the sense of
Lyapunov even in the presence of parameters uncertainties. Simulations were also performed for different
initial conditions and in all cases the control was successfully achieved.
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9. ADDITIONAL READING SECTION
Chaos in aerospace and aeronautical systems
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Macau, E. E. N. (2008). Low-Thrust Chaotic Based Transfer from the Earth to a Halo Orbit. Journal of
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supersonic flow. Acta Mechanica Solida Sinica, 21(5), 441–448.
Chaos in other engineering systems
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Lixia, J., & Wanxiang, L. (2008). Chaotic Vibration of a Nonlinear Quarter-Vehicle Model. In IEEE
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Engineering, 88(1), 35–44.
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Zaher, A. a. (2008). A Nonlinear Controller Design for Permanent Magnet Motors using a
Synchronization-based Technique Inspired from the Lorenz System. Chaos, 18(1), 013111.
Control and synchronization of chaotic systems
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10. KEY TERMS AND DEFINITIONS
Chaotic: Represents the type of motion that is highly sensitive to changes in the initial conditions, that is
extremely sensitive to parameter changes, and that its behaviour is unpredictable for medium/longterm. Two trajectories initiated with slightly different initial conditions diverge exponentially over
the time.
Limit cycle: An isolated closed trajectory in the phase space. The term isolated means that the
neighbouring trajectories are not closed.
Lyapunov exponents: Numbers that measure the rate of exponential attraction or separation over the
time of two adjacent trajectories started with different initial conditions in the phase space. For a
dynamical system with bounded trajectories, a positive Lyapunov exponent indicates the existence of
a chaotic motion and if two or more exponents are positive the system is said to be hyperchaotic.
Melnikov function: A function that measures the distance between a stable manifold and an unstable
manifold associated with a saddle point of a given Poincaré section resulting from a continuous flow.
According to the Melnikov theory when the two manifolds intersect the function has a simple zero
indicating the existence of chaos,
Phase space: For dynamical systems governed by a set of first order differential equations is a space
whose coordinates are the state variables or some components of the state vector. It is used to
represent the system behaviour and a point in the phase space defines a potential state of the system.
Poincaré section: A sequence of points in the phase space generated by the intersection of a continuous
flow with a given surface/section.
Strange/Chaotic attractor: Refers to an attractor set in the phase space with a fractal dimension in which
the motions of the trajectories are chaotic.
Trajectory/Orbit: A path (sequence of points) generated by a continuous or discrete system in its phase
space.
Underactuated systems: Systems in which the number of control variables is less than the number of
degrees of freedom.