Adv Syst Sci Appl 2017; 4:1–13
Published online at http://ijassa.ipu.ru/ojs/ijassa/article/view/481

Predictive Feedback Control Method for Stabilization of
Continuous Time Systems

Lina A. Shalby1,2

1Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region, Russia
2Faculty of Women, Ain Shams University, Cairo, Egypt

Abstract: Recent researchers have studied the chaotic phenomenon and introduced methods to
control chaos. One of these methods is the Predictive Feedback Control (PFC). PFC method has
been used to stabilize only the discrete chaotic maps. In this paper, we generalize the PFC method
by extending it to stabilize continuous time systems. The numerical simulations are carried out
to solve the system using Euler method for its simplicity, and then the PFC method is applied to
the discretized system. The controlled trajectory converges to an unstable equilibrium point via
small control action. The stability analysis is shown compared to that of the continuous system in
details. Also the choice of the controlling input of the PFC method and its properties are discussed.
Lorenz system and Rössler system are the well known chosen chaotic examples. The PFC method
is applied to each of them, showing stability at each of their equilibrium point. With a very small
control, the behaviour of the system is completely changed from chaos to stable.

Keywords: Continuous time systems, chaos control, Euler method, Predictive feedback control,
Unstable equilibrium point

1. INTRODUCTION

Chaos theory is one of the recent fields of study that attracts the interest of scientists. Two
most important properties of chaos are the high sensitivity of trajectories to initial conditions
and to parameter values. In the last decades, many researchers concerned with how to control
chaotic phenomena depending on these two properties. They aimed to bring a trajectory to a
small neighborhood of a desired location in the chaotic attractor, using a small perturbation.
Pioneering work Ott-Grebogi-Yorke [12] proposed OGY method for stabilizing chaos. It
could only be applied to discrete data, so continuous time systems should first be made
discrete time by using the Poincare map. Then it was followed by many other publications
adjusting and extending the OGY method as in [6] and [7], see also survey [1]. Pyragas
[14,15] proposed the delayed feedback control (DFC) to stabilize the unstable periodic orbits
(UPO). The control input was given as the multiplication of a gain with the difference between
current system state and a state with the time delay as it is discussed in [4]. DFC method is
firstly constructed for continuous time systems and extended to the discrete-time case in [18].
Many applications, such as in [3, 10], were controlled by using DFC method. Also, many
developments for DFC method were performed, see [8, 11]. One of the recent surveys on
DFC is presented in [16].

Ushio [19] introduced the idea of predicting control in his paper and it was extended
in [2] to stabilize continuous time systems using predictive-based control method. Predictive

∗Corresponding author: lina.khamis@gmail.com, Lina.Khamis@women.asu.edu.eg

http://ijassa.ipu.ru/ojs/ijassa/article/view/481


2 LINA A. SHALBY

Feedback Control (PFC) [13] is an easily implemented method to stabilize unknown UPOs
in discrete time dynamical systems. In PFC, the control input was given as the multiplication
of a gain with the difference between two predicted future system states. PFC depends on
applying small controls that completely change the nature of system’s behavior and devotes
its attention to the most important problem of stabilizing or controlling chaos. Detailed survey
on existing approaches and methods for chaos control can be found in the Handbook of Chaos
Control [17].

In the next section of the paper, we extend PFC to stabilize chaotic continuous time
dynamical systems solved numerically by Euler method and the PFC method is analyzed.
In section 3, the numerical simulations results are given for the most popular examples of
chaos: Lorenz system and Rössler system.

2. PREDICTIVE FEEDBACK CONTROL METHOD

We start with introducing the PFC method [13] for discrete-time system which depends on
the system state and a control action u(x) as follows

xn+1 = F (xn)− u(xn), (2.1)

where F, u : RN → RN and u(x) = E(F p+2(x)− F p+1(x)), for a chosen control gain matrix
E and any arbitrary forward predicted state p (p > 0 ), by denoting F p+1(x) = F (F p(x)),
F 1(x) = F (x) — prediction of the map F for p+ 1 iterations forward. The idea is to
choose E to stabilize unstable orbits or unstable equilibria points via small control action
u(x). Indeed state x and control u are of the same dimension N , and this strongly limits
the application of the proposed technique to real-life problems. However there are various
situations where such controls are applicable. For instance if one deals with classical
analytical examples (e.g. Rössler and Lorenz systems) it is of interest to deal with their fixed
points and orbits which are unstable. Then PFC allows to stabilize such points and orbits. And
the main peculiarity of the approach is the arbitrary small control action required. Suppose
N = 1, x∗ is an unstable equilibria point point of F (x), that is x∗ = F (x∗), µ = |F ′(x∗)| > 1.
Then for E = 1

µp+1(µ−1) , iterations of (1) converge to x∗. The most important fact is obtaining
control u(x) small enough. Indeed, if we deal with chaos control, iterations F p(x) remain
bounded, while E is small for p large, because µ > 1. It is effective to choose p large enough
(because larger is p, smaller is the control action). However, large predictive horizons imply
computational error. Thus the choice of p is some trade-of. In [13] the result is extended for
N > 1 and for unstable orbits (i.e. for x∗ such that x∗ = F k(x∗) with k > 1). Similar results
are available for µ known approximately.

The goal of the present paper is the application of the above results for continuous-time
systems. Let us consider a system of nonlinear autonomous differential equations

ẋ = f(x), (2.2)

where x ∈ RN , and f is assumed to be differentiable. An equilibrium point x∗ necessarily
satisfies f(x∗) = 0. The following result on asymptotic stability is standard (see e.g. [5]).
The equilibrium point x = x∗ of ẋ = f(x) is asymptotically stable if all eigenvalues of the
Jacobian matrix A = f ′(x∗) satisfy Re(λi) < 0.

From now on, we reduce a continuous time system to a discrete time map. This map
is created from sampling the flow at discrete times tn = t0 + nh, n = 0, 1, 2, . . . , where the
sampling interval h can be chosen on the basis of convenience. Thus, a continuous time
trajectory x(t) yields a discrete time trajectory xn = x(tn). By using Euler method, the
simplest numerical method to solve a differential equation, the system will be discretized
easily to take the form

xn+1 = xn + hf(xn) = F (xn), (2.3)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 3

where F (x) = x+ hf(x). Equation (2.3) is the simplest discrete form corresponding to the
continuous time system. Lemma 2.1 highlights the relation between the discretized function
F and the continuous system function f . Let µi denote the eigenvalues of the Jacobian matrix
M = F ′(x∗) = I + hf ′(x∗), f(x∗) = 0, while λi are eigenvalues of A = f ′(x∗).

Lemma 2.1:
F (x) satisfies the following:
(i) F(0) = 0, and F (x∗) = x∗ + hf(x∗) = x∗.
(ii) Let Re(λi) = ui and Im(λi) = νi, if ui < 0 and 0 < h < 2|ui|

u2i+ν
2
i
, then

|µi| = |1 + hui| < 1, and if ui ≥ 0, then |µi| ≥ 1.

Lemma 2.1 shows that the discrete form for the continuous system has the same
equilibrium point, and unstable equilibrium points of the continuous system imply instability
of the discrete system. Thus for the discretized system, the predictive feedback control
method in [13] can be used to stabilize the system as illustrated for the vector case. Thus
we arrive to the following stabilization scheme for continuous system (2), where the matrix
E could be considered to be the diagonal with entries εi. The choice of εi depends on the value
of the eigenvalues µi of the Jacobian matrix M , which could be real or complex. Firstly, for
| µi |< 1, we take εi = 0. Secondly, if | µi |≥ 1 and µi ∈ R , then εi = 1

µp+1
i (µi−1)

. Thirdly, if

| µi |≥ 1 and µi ∈ C, such that µi = ai + bij, j =
√
−1, then the control input εi will be a

2× 2 matrix: εi = D−p−1(D − I2)−1 , where D =

[
ai bi
−bi ai

]
. We restrict our analysis with

fixed points, because orbits are fixed points of the iterated maps. The next theorem and its
proof illustrate local stability of the system for these choice of the control gain.

Theorem 2.1:
Let f ∈ C1, x ∈ RN and assume x∗ is an unstable equilibrium point of continuous system
(2.2) and its Jacobian matrix M is not Hurwitz. Then x∗ is stable fixed point of the predictive
feedback control equation (2.1) after discretizing system (2.2) numerically by using Euler
method (2.3) provided h is small enough.

Proof
At x = x∗, matrix A has eigenvalues λi, with ui = Re(λi) > 0 for some i. Suppose that
F (x) is h-step sized Euler approximation of Equation (2.2), given by Equation (2.3), then the
Jacobian matrix M has the eigenvalues µi = 1 + hλi. By the chain rule, the Jacobian matrix
of F p(x) is associated with the eigenvalues µpi . Let J is the Jacobian matrix of the right hand
side in Equation (2.1) at x = x∗, which will have the form, J =M − E {Mp+2 −Mp+1} =
M − EMp+1 {M − I},
and J has eigenvalues ηi , i = 1, ..., N . Then | ηi |=| µi − δ {µi − 1} |< 1∀i, for the
following cases of µi :
For all | µi |< 1, εi = 0 and ηi = µi.
If | µi |≥ 1 and µi ∈ R, for εi =

δh|λi|
µp+1
i (µi−1)

= δ

µp+1
i

, 1 < δ < 1
h|λi| , then | ηi |=| µi − δ(µi −

1) |=| 1 + hλi − δhλi |=| 1− (δ − 1)hλi |< 1.
While if | µi |≥ 1 and µi ∈ C, for εi = δh | λi | (D−p−1(D − I2)−1), 1 < δ < 1

h|λi| where

D =

[
ai bi
−bi ai

]
has the same eigenvalues µi and Dp has the eigenvalues µpi . Then | ηi |=|

µi − δh|λi|
µp+1
i (µi−1)

µp+1
i (µi − 1) |=| 1 + hλi − δh | λi ||< 1.

Hence the system is stable since the eigenvalues of J are less than unity.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



4 LINA A. SHALBY

The next section illustrates the examples of chaotic systems and their stabilization
achievement after adding the control term. The examples are clearly showing easy
implementation and the efficiency of the PFC method.

3. EXAMPLES

In this section, we provide two examples of the PFC method applied to classical chaotic
system. Lorenz model is 3D system example, and Rössler system is 4D example.

3.1. Lorenz Model
Lorenz [9] was the first who introduced the strange attractor notion and coined the term
“butterfly effect”. Lorenz system is described by differential equations

ẋ = −σ(x− y)
ẏ = −xz + rx− y

ż = xy − kz
(3.4)

Equation (3.4) has two nonlinear terms, and exhibits both periodic and chaotic motion
depending upon the values of the control parameters σ, r and k. By using Euler method the
system will be discretized to the form

xn+1 = xn − hσ(xn − yn)
yn+1 = yn + h(−xnzn + rxn − yn)

zn+1 = zn + h(xnyn − kzn)
(3.5)

where h is the step size for discretization of time t. The stability analysis of Equation (2.1)
can be performed by studying the discrete 3D function corresponding to it,

F (x, y, z) =

x− hσ(x− y)y + h(−xz + rx− y)
z + h(xy − kz).

(3.6)

The system (3.4) has three fixed points s1 = (−b
√
r − 1,−b

√
r − 1, r − 1), s2 =

(0, 0, 0) and s3 = (b
√
r − 1, b

√
r − 1, r − 1), which are the same as the equilibrium points

of the continuous system (we denote 3D vector (x, y, z) as s). Set σ = 10 and k = 2.67,
and make r the adjustable control parameter. Varying the values of r reveals a critical
value at rc = 24.74. Below rc the system decays to steady, non-oscillating, state. Once
r increases beyond rc, the continuous oscillatory behavior occurs and the system shows
aperiodic behavior which Lorenz called deterministic non-periodic flow which refer to
chaos. For r = 28, the chaotic case, the equilibrium points are s1 = (−8.4906,−8.4906, 27),
s2 = (0, 0, 0) and s3 = (8.4906, 8.4906, 27) and the eigenvalues corresponding to them are
λs1 = λs3 = {−13.8569, 0.0934± 10.2001j} and λs2 = {−22.8277, 11.8277,−2.67}. Let
us use the step size h to be a small value equal to 0.001 which satisfies the condition due
to the complex value of λs1 (i.e h < 2×0.0934

(0.0934)2+(10.2001)2
= 0.00179), and then the solution of

the system is chaotic as shown in Fig. 3.1.

For the discrete form, the eigenvalues are µs1 = µs3 = {0.9861, 1.00009± 0.0102j} and
µs2 = {0.99733, 1.0118, 0.9771}. The complex eigenvalues have the absolute value 1.00014.
Let us choose p = 5, the initial condition to be (5, 5, 5), and step size h = 0.001. Starting with
stabilizing the system at the origin, at which ε1 = ε1 = 0 for | µ(1,3)

s2 |< 1, and otherwise we

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 5

Fig. 3.1. Phase space plot for Lorenz system.

choose δ = 2, ε2 = 2

(µ
(2)
s2 )p+1

. The matrix E is taken to be in the form E =

[
0 0 0
0 1.8638 0
0 0 0

]
and the first iterate of the decaying additive control term is −0.2040.

Fig. 3.2. The distance between s and 0 before and after stabilization.

Fig. 3.3. Phase space plot of the controlled trajectory.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



6 LINA A. SHALBY

Fig. 3.2 illustrates the difference between the equilibrium (0, 0, 0) and the values of x-
trajectory, y-trajectory and z-trajectory respectively, before stabilizing the system in blue
color compared to those after applying the controlling term to the system that is coloured
by red. Fig. 3.3 is the phase space plot of the stabilized system. Both figures show that the
system is stable and all variables time series are converging to the origin starting from initial
point (5, 5, 5).

In case of the complex eigenvalues, µ
(1)
s1 = µ

(1)
s2 = 0.9861 and µ

(2)
s1 = 1.00009±

0.0102j, ε1 = 0, and D =

[
1.00009 0.01020
−0.01020 1.00009

]
,ε2,3 = δh | λ2,3 | D−p−1(D − I2)−1 =[

0.01030 −0.1978
0.1978 0.01030

]
, and E =

[
0 0 0
0 0.0103 −0.1978
0 0.1978 0.0103

]
.

Fig. 3.4. Comparison between the controlled and chaotic trajectories for the equilibrium point
(-8.4906, -8.4906, 27).

Fig. 3.5. Phase space plot for the controlled trajectory for the equilibrium point (-8.4906, -8.4906, 27).

In Fig. 3.4 and Fig. 3.5, the equilibrium point (−8.4906,−8.4906, 27) is stable. To
stabilize the system at (8.4906, 8.4906, 27), the negative sign of the imaginary part of the

complex eigenvalue is reversed, so that the matrix D will be
[
1.00009 −0.01020
0.01020 1.00009

]
.

Both Fig. 3.6 and Fig. 3.7 show the slow convergence of the trajectory after stabilization
due to the point (8.4906, 8.4906, 27). It is important and interesting to test the effect of
different initial conditions. For 50 random initial conditions, the convergence has been
tested. At t = 10, the trajectories succeeded to reach the fixed points (0, 0, 0) (see Fig. 3.8)

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 7

Fig. 3.6. Comparison between the controlled and chaotic trajectories with respect to each dimension with the
equilibrium point (8.4906, 8.4906, 27).

Fig. 3.7. Phase space plot for the controlled trajectory due to the equilibrium point (8.4906, 8.4906, 27).

and (8.4906, 8.4906, 27) (as shown in Fig. 3.9) for all initial points x0, y0 ∈ [−10, 10] and
z0 ∈ [0, 30].

All the statements rely on local analysis. But in some cases, there was an effect similar to
global stabilization, as it is seen in Fig. 3.8 and Fig. 3.9. This fact is a motivation for research
of global behavior in future works.

3.2. Rössler Model
Rössler model of 4D phase space is one of the most famous hyperchaos models. For x ∈ R4,
its defining equations are

ẋ = −y − z
ẏ = x+ αy + w

ż = β + xz
ẇ = −γz + τw

(3.7)

The chaotic behavior of the system (3.7) is observed for the parameter values
α = 0.25, β = 3, γ = 0.5 and τ = 0.05. In this case, the equilibrium points are
eq1 = (5.40833, 0.55470,
−0.55470,−5.54700) and eq2 = (−5.40833,−0.55470, 0.55470, 5.54700), which are
associated with the eigenvalues λeq1 = {5.504039, 0.103929, 0.0501791± 0.971052j}
(the step size h must be less than 0.1061) and λeq2 =
{0.101890, 0.0493731± 0.9986873j,−5.308963} (the step size h must be
less than 0.0987) respectively. By solving the system with Euler method,

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



8 LINA A. SHALBY

Fig. 3.8. The stability is reachable at (x(10), y(10), z(10)) = (0, 0, 0) for 50 random initial conditions.

Fig. 3.9. The stability is reachable at (x(10), y(10), z(10)) = (8.4906, 8.4906, 27) for 50 random initial
conditions.

adjusting the step size h = 0.001(suitable for stability conditions stated in
Lemma 2.1), the continuous system translated to the discrete form with
eigenvalues µeq1 = {1.005504, 1.000103929, 1.00005017± 0.00097105j} and µeq2 =
{1.000101890, 1.000493731± 0.0009986873j, 0.9946910}. Let us use the initial condition
(−10,−6, 0, 10), then the system (3.7) is hyperchaos as shown in Fig. 3.10 and Fig. 3.11 for
the phase space xyw and x− y plot, respectively.

By setting p = 3, δ1,2 = 1.5 and δ3,4 = 5 for ε1 =
δ1

(µ
(1)
eq1)

4
= 1.46743, ε2 = δ2

(µ
(2)
eq2)

4
=

1.49937, and ε3,4 = δ3,4 × h | λ(3)eq1 | ×D−4(D − I2)−1 =
[
0.01243 0.26011
−0.26011 0.01243

]
, then

Eeq1 =

 0.01243 0.26011 0 0
−0.26011 0.01243 0 0

0 0 1.46743 0
0 0 0 1.49937

. While δ1 = 1.5 and δ2,3 = 40 for

ε1 =
δ1

(µ
(1)
eq2)

4
= 1.49939 , ε2,3 = δ2,3 × h | λ(2)eq2 | ×D−4(D − I2)−1 =

[
0.09053 1.99260
−1.99260 0.09053

]
,

then Eeq2 =

 0.09053 1.99260 0 0
−1.99260 0.09053 0 0

0 0 0 0
0 0 0 1.49939

.

The first iterate of the control term values are -0.00031, -0.00158, 0.00419, and 0.00074
to stabilize the system at eq1. And its first iterate values to stabilize the system at eq2 are
-0.00237, -0.01207, 0, and 0.00074. The control term values are small negligible in the real
life applications but they are enough to change the behavior of the system from ergodicity to
be stable. Fig. 3.12 and Fig. 3.13 are showing the controlled phase space xyw due to eq1 and
eq2, respectively. And Fig. 3.14-3.17 are the trajectories convergence of x, y, z and w to the
equilibrium points compared to the uncontrolled trajectories, respectively.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 9

Fig. 3.10. The strange attractors of Rössler hyperchaos system - xyw phase space plot.

Fig. 3.11. x− y plot for Rössler hyperchaos system.

For each example, the PFC method is successfully stabilized all the equilibrium points.
It is also important to discuss the relation between the continuous system and its discretized
form numerically, as they are difficult to be solved analytically. At this point, a question may
arise. What is the continuous system that has stable attractor while its numerical solution is
chaotic?

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



10 LINA A. SHALBY

Fig. 3.12. The xyw phase space after control at eq1.

Fig. 3.13. The xyw phase space after control at eq2.

Fig. 3.14. The x trajectory behavior before control blue colored trajectory and after control red colored trajectory
compared to eq1 and eq2 in green respectively.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 11

Fig. 3.15. The y trajectory behavior before control blue colored trajectory and after control red colored trajectory
compared to eq1 and eq2 in green respectively.

Fig. 3.16. The z trajectory behavior before control blue colored trajectory and after control red colored trajectory
compared to eq1 and eq2 in green respectively.

Fig. 3.17. The w trajectory behavior before control blue colored trajectory and after control red colored
trajectory compared to eq1 and eq2 in green respectively.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



12 LINA A. SHALBY

4. CONCLUSION

In this paper, we have extended PFC method to stabilize chaotic continuous time systems
by discretizing the system numerically using Euler method, and then applying the PFC
method. The PFC method is used by adding small control term to the iterative solution of the
system, this term is the multiplication of a controlling matrix by the difference between two
consecutive predicted iterates. We have also discussed the stability analysis of the continuous
system compared to its numerical discrete form, and the choice of the control matrix due to
the eigenvalues of the discretized system type either real or complex, and we have proved that
the method is stable. The method has been applied to the most popular systems, Lorenz 3D
system and Rössler hyperchaos 4D system. Their trajectories have been controlled to each
equilibrium point after a few iterations. The extended PFC method is easy implemented and
highly efficient method for all chaotic systems to be stabilized, as we have generalized PFC
method for all chaotic systems either discrete or continuous time system.

ACKNOWLEDGEMENTS

It is my pleasure to have the supervision of Prof. B. T. Polyak. And I would like to express
my gratitude to him, for the patient guidance, support and advice he has provided me in this
work. Also we would like to express our thankful to the editor and anonymous reviewers of
the journal, for their great efforts and valuable revision.

REFERENCES

1. Andrievskii, B. R. & Fradkov, A. L. (2003) Control of Chaos: Methods and
Applications. I. Methods. Automation and remote control, 64 (5), 673–713.

2. Boukabou, A. & Mansouri, N. (2008) Predictive Control of Continuous Chaotic
Systems. Int. J. Bifurcation Chaos, 18 (2), 587–592.

3. Celka, P. (1994) Experimantal verification of Pyragass Chaos Control method applied
to Chuas circuit. Int. J. Biforcation Chaos Appl. Sci. Eng., 4 (6), 29–36.

4. De Sousa Vieira, M. & Lichtenberg, AJ. (1996) Controlling chaos using nonlinear
feedback with delay. Phys. Rev. E, 54 (2), 1200–1207.

5. Khalil, H. K.(2002) Nonlinear systems. Upper Saddle River, NJ: Prentice Hall.
6. Kostelich, E. J., Grebogi, C., Ott, E. & Yorke J. A.(1993) Higher-dimensional targeting.

Phys. Rev. E, 47 (1), 305–310.
7. Lai, Y.-C., Ding, M. & Grebogi, C. (1993) Controlling hamiltonian chaos. Phys. Rev. E,

47 (1), 86–92.
8. Leonov, G. A., Zvyagintseva, K. A. & Kuznetsova, O. A. (2016) Pyragas stabilization of

discrete systems via delayed feedback with periodic control gain. IFAC-PapersOnLine,
49 (14), 56–61.

9. Lorenz, E.(1963) Deterministic non-periodic flow. Journal of the Atmospheric Sciences,
20 (2), 130–141.

10. Mitsubori, K. & Aihara, K. (2002) Delayed–feedback control of chaotic roll motion of a
flooded ship in waves. In Proceedings of the Royal Society of London A: Mathematical,
Physical and Engineering Sciences, 458 (2027), 2801-2813.

11. Morgul, O. (2003) On the Stability of Delayed Feedback Controllers. Phys. Lett. A,
341(4), 278–285.

12. Ott, E., Grebogi, C. & Yorke, J. A.(1990) Controlling Chaos. Phys. Rev. Lett., 64(11),
1196–1199.

13. Polyak, B. T. (2005) Stabilizing Chaos with Predictive Control. Automation and Remote
Control, 66(11), 1791–1804.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)



PFCM FOR STABILIZATION OF CONTINUOUS TIME SYSTEMS 13

14. Pyragas, K. (1992) Continuous Control of Chaos by Self-Controlling Feedback. Phys.
Lett. A, 170 (6), 421-428.

15. Pyragas, K. (1995) Control of Chaos via Extended Delay Feedback. Phys. Lett. A,
206(5-6), 323-330.

16. Pyragas, K. (2006) Delayed feedback control of chaos. Philosophical Transactions of
the Royal Society of London A: Mathematical, Physical and Engineering Sciences,
364(1846), 2309–2334.

17. Schöll, E. & Schuster, H. G.(2008) Handbook of chaos control, John Wiley & Sons.
18. Ushio, T. (1996) Limitation of Delayed Feedback Control in Nonlinear Discrete-Time

Systems. IEEE Trans. Circ. Syst., 43(9), 815–816.
19. Ushio, T., and Yamamoto, S. (1999) Prediction-based Control of Chaos. Phys. Lett. A,

264(1), 30–35.

Copyright c© 2017 ASSA. Adv Syst Sci Appl (2017)


	Introduction
	Predictive Feedback Control Method
	Examples
	Lorenz Model
	Rössler Model

	Conclusion