1. Introduction
Since Lorenz found the famous Lorenz model describing the dynamics of the atmosphere [
1], much effort has been devoted to the study of nonlinear chaotic systems. Many fundamental properties can be found in chaotic systems, such as broad spectrums of Fourier transform, strange attractors, excessive sensitivity to initial conditions (so-called butterfly effect), and fractal properties of the state responses [
2]. It has also been mentioned that the Lorenz model can depict many different engineering systems, for example, laser devices, the disk dynamics and some issues concerned with convection [
3,
4]. Due to extensive applications of chaotic systems in solving engineering problems, controlling the complex dynamics of chaotic systems has emerged as an attractive issue for engineering applications and many profound control schemes as well methodologies can be found in the literature [
2,
5,
6,
7,
8,
9,
10,
11,
12]. Consequently, many different effective approaches have been presented to cope with the problems of control and stabilization for various classes of chaotic systems, for example, sliding mode control [
5,
6,
7,
8], backstepping design [
9,
10], optimal control [
11,
12], etc.
In 2017, in order to make more precision models to describe the atmosphere, the authors in [
4] proposed a generalized 4D Lorenz–Stenflo system with six parameters. In accordance with the detailed discussion in [
13], this proposed generalized Lorenz–Stenflo system has three positive Lyapunov exponents and exhibits interesting and complex chaotic dynamics, since the generalized Lorenz–Stenflo system is hyper-chaotic in behavior and it is not easy to control, especially when this controlled system is under the influence of external disturbances and nonlinear inputs. In [
14], the authors introduced four control inputs to achieve the hyper-chaos suppression. However, the control design cannot apply to the systems which are subjected to external disturbances or nonlinear inputs. Generally, the nonlinearity in the control implementation is highly undesired but always unavoidable due to the physical limitation. If the input nonlinearity is not well taken into account, then it leads system to failure [
15,
16]. In [
17], the authors proposed fuzzy control schemes to control the state variables of the Lorenz–Stenflo chaotic systems to its equilibrium point. By selecting specified parameters, the adaptive controlled system can approach the equilibrium point with high precision. However, the uncertainties considered in [
17] need to be dependent on the system states and no clear formula is presented for estimating the control performance. Motivated as mentioned before, the goal of the research is to propose a robust controller to suppress the chaotic behaviors for 4D generalized Lorenz–Stenflo systems subjected to matched/mismatched uncertainties and input nonlinearity. Contrary to previous works [
5,
6,
7,
8,
9,
10,
11,
12,
14] for chaos suppression control, we will consider the input nonlinearity which will always exist in circuit realization for control input. Furthermore, a new control concept called rippling control is introduced for control design. In the rippling control, only certain states of systems are first controlled and, when specified states are suppressed and stabilized, the other states will also be suppressed in the manner of ripple. Thus, the ripple control can not only effectively reduce the number of control inputs but also achieve chaos suppression. Reducing the number of control inputs has remarkable significance in decreasing the complexity of controller realization. Therefore, we will utilize a rippling control method in this research such that only a single control input is used for completing suppression of hyper-chaotic behavior in 4D generalized Lorenz–Stenflo systems even with the undesired input nonlinearity.
The rest of this paper is given as follows. In
Section 2, we describe the 4D generalized Lorenz–Stenflo system and formulate the suppression problem of chaos dynamics. In
Section 3, integrating the rippling control concept with the sliding mode control, a PI switching surface is designed firstly and the bounds of the controlled chaotic dynamics in the sliding manifold are derived. In
Section 4, a rippling sliding mode controller is implemented to achieve the hitting in spite of input with nonlinearity and external disturbances. A procedure for the rippling controller design is proposed. In
Section 5, we give the illustrative examples with matched and unmatched conditions, respectively. Finally, conclusions are presented in
Section 6.
Notations: In this paper, represents the n-dimensional Euclidean space, denotes the set of all real n by m matrices, and stands for an m by m identity matrix. denotes the Euclidean norm and the induced norm for a matrix W and a vector W, respectively. represents the absolute value of and denotes the sign function of s and we have if s > 0; if s = 0; if s < 0. is a diagonal matrix.
2. System and Problem Descriptions
The generalized Lorenz–Stenflo system is described as
where
, are state variables and
are the system parameters with positive values. System (1) exhibits hyper-chaotic strange attractors, as shown in
Figure 1 when
[
4].
A 4D generalized Lorenz–Stenflo system with mismatched uncertainties can be described as
Generally, the considered uncertainties
in (2) are bounded by
where
are given. For effectively controlling the uncertain system (2), we introduce a controller
to the system (2). After introducing this single input subjected to input nonlinearity, the controlled system can be expressed by
where
denotes the control input,
is the control input with nonlinearity. The nonlinear input
satisfies
,
and is bounded in the sector
, the
and
are the known positive constants. Therefore, it can be formulated as
The nonlinear control input
is shown in
Figure 2.
The goal of this paper is to propose a single rippling SMC to suppress the chaos behavior in the uncertain 4D Lorenz–Stenflo system to zero or predictable bounds even with mismatched uncertainties and input nonlinearity, i.e.,
where
are the upper bounds estimated which will depend on external uncertainties.
By utilizing the sliding mode control technology to achieve our control goal, two major problems need to be solved well. First, we need to decide a proper switching surface for the controlled system (4) to ensure that the dynamics in the sliding manifold have predicted bounds with , even under the influence of external uncertainties. Next, we need to determine a sliding mode controller such that the trajectory of controlled system (4) can be driven to the switching surface even with the nonlinearity of gain reduction tolerance in the control input.
3. PI Switching Function and Estimated State Bounds in the Sliding Manifold
In this section, we will propose the design procedures of the controller. To reduce the number of control inputs, we introduce the concept of the rippling control and only the first three states
of the system in (4) are controlled. When the specified states are suppressed and stabilized, the fourth state
will also be suppressed in the manner of ripple. Therefore, we first select a proper switching surface which is only relative in the first three states
of system in (4) to ensure the control performance in the sliding manifold; then, a robust SMC is developed to guarantee the hitting of the sliding mode manifold even with matched/unmatched uncertainties and input nonlinearity. The proportional-integral (PI) switching surface is selected as
where
,
,
is a designed gain matrix that can be easily determined later. Assuming that the system operates in the sliding mode when
, according to the sliding mode control theorem, the equivalent control law can be obtained due to the facts of
and
. Considering (7) and (4), we have
From (8), if the controlled system is in the sliding manifold, we have
and the equivalent control law can be obtained as
Substituting (9) into (4) yields
Rearranging the differential equations for the first three states in (10), we have
where
Since (A, B) is controllable, we can always select a specified gain matrix K by using any pole assignment method such that eigenvalues of matrix are all different and satisfy .
Solving (11), one has for
Now, selecting to satisfy , is the independent eigenvector corresponding to eigenvalue of matrix .
Since
, one has
and the solution of
can be obtained as
where
is
-row of
.
The bounds of
, for
, can be estimated by
The definite integral of
in (15) can be represented by the limit of the Riemann sum [
18] described as below:
where
and
.
By substituting (17) into (15), we have
Since
and
, we have the bounds
of
as
Similarly, from (2), since
, we can calculate
as
Remark 1. For the system with, according to (19) and (20), we can conclude that when system (4) enters the sliding manifold, all controlled states will be forced to zero, i.e.,, even with the matched uncertainty.
Although we have estimated the state bounds of controlled dynamics in the sliding manifold, we still need to determine a control input such that the controlled system can be driven to the switching surface and the existence of the sliding motion can be guaranteed even with the gain reduction tolerance in the control input.
4. Robust Controller Design for Sliding Motion
In order to design the SMC scheme, the hitting condition is given below.
Lemma 1. The trajectory of controlled systems can converge to the sliding mode (7), i.e.,,
if the following hitting condition is satisfied: Proof of Lemma 1. Let , for all , be the Lyapunov function. From the Lyapunov stability theory, condition (21) implies and as well as that the switching function can converge to zero. □
To achieve the reaching condition indicated in Lemma 1, the control input subjected to input nonlinearity is proposed as
In the following, we will prove that the proposed scheme (19) can drive the uncertain system (4) with input nonlinearity onto the sliding mode .
Theorem 1. Considering that the uncertain system (4) with input nonlinearity, let the controlbe given as (22), then, the system trajectory asymptotically converges to the sliding manifold.
Proof of Theorem 1. Substituting (4) and the control (22) into the derivative
, we obtain
Furthermore, by multiplying
to (24) and using the fact of
, we have
From (25), we can further derive
Since
, combined with (26), (23) yields
Thus, according to Lemma 1, one can conclude that the system trajectory asymptotically converges to the sliding manifold . □
Remark 2. From the theoretical point of view, the proposed control input (22) may result in chattering due to the discontinuous sign function. A simple approach to solve this problem is to modify the controller (22) aswhere is a sufficiently small positive constant. Obviously, the continuous controller (28) with a sufficiently small value of can approach the discontinuous controller (22) very well. We can systematize the design procedure for robust chaos synchronization as follows.
Remark 3. According to the above discussion, we can systematize the design procedure for robust chaos suppression in generalized 4D Lorenz–Stenflo systems as follows.
Step 1: Select the gain matrix to guarantee that the eigenvalues of matrix in (11) are all different and satisfy , which will ensure a stable sliding manifold.
Step 2: Construct the switching function in (7).
Step 3: Find independent eigenvectors corresponding to eigenvalue of matrix and construct the transform matrix .
Step4: Obtain the predicted error bounds by using (19) and (20).
Step5: Construct the SMC from (22) or (28).
5. Numerical Simulations
In this example, in the generalized 4D Lorenz–Stenflo system (1), the six parameters are selected as
. We can easily check that the pair
is controllable and, according to Step 1 in Remark 3, we can easily select the gain matrix
such that
to result in a stable sliding motion. According to (7), the switching function
can be obtained as
and the transform matrix
For simulation, we define the nonlinear input as
. Obviously, we can obtain
and
. The sliding mode controller can be constructed as
with
.
In the following, numerical experiments including matched and unmatched conditions of uncertainties are given to verify the control performance of the present SMC design.
5.1. Robust Control with Matched Uncertainty and Input Nonlinearity
The unmatched uncertainties are set as
. The matched uncertainty is given as
. In this simulation, we enable the control input at
s. According to (19) and (20), we can understand that the controlled system states will be exactly driven to zero.
Figure 3 and
Figure 4 show the simulation results with initial condition
.
Figure 3 shows the state responses under the proposed controller (30).
Figure 4a,b, respectively, show the corresponding
for the controlled 4D Lorenz–Stenflo system and the control input for
s. By observing the simulation results, it reveals that the controlled trajectory of the system hits
at
s. and the states approach to zero after
, i.e., the system state is exactly forced to zero and the proposed SMC (30) is robust to the matched uncertainty and input nonlinearity.
5.2. Robust Control with Mismatch Uncertainties and Input Nonlinearity
Here, we consider the control performance under the influence of mismatch external uncertainties . The control input becomes active at s.
Obviously, we have . Furthermore, by Equations (19) and (20), we can calculate that the controlled states are bounded by .
The initial condition
is used in this simulation. The state responses are shown in
Figure 5. The detailed state responses for
s are shown in
Figure 6 to get a clearer view. According to
Figure 6, it reveals that the controlled states are suppressed and bounded by
, as we estimate with (19) and (20).
Figure 7a,b, respectively, show the corresponding
and the control input for
s. From
Figure 7b, it can be observed that the chattering phenomenon can disappear when
s.
6. Conclusions
A robust chaos suppression control for 4D generalized Lorenz–Stenflo systems subject to matched/mismatched uncertainties and input nonlinearity is proposed in this paper. A new rippling sliding mode controller has been developed to regulate the state vector. According to the mathematical analysis and simulation results, we can observe that the proposed sliding mode controller is effective for controlling uncertain 4D generalized Lorenz–Stenflo systems even with matched/mismatched uncertainties as well as input nonlinearity. Moreover, the bounds of controlled system states have also been well discussed under the effect of mismatch uncertainties. The numerical simulation results demonstrate the robustness and validity of the proposed chaos suppression controller. In the near future, our main work is to extend the proposed rippling sliding mode control to suppress the stochastic chaos systems with multiplicative noises.