A new sliding mode control model of discrete chaotic systems based on multimodal function series coupling is proposed to overcome the shortcomings of the standard PSO algorithm in multimodal function optimization. Firstly, a series coupled PSO algorithm (PP algorithm) based on multimodal function is constructed, which is optimized by multipeak solution on the basis of the standard PSO algorithm. Secondly, the improved PSO algorithm is applied to search all the extreme points in the feasible domain. Thirdly, the Powell method is used to perform the local optimization of the search results, and the newly generated extreme points are added to the extreme point database according to the same peak judgment operator. Finally, the long training time of PP algorithm can be overcome by the characteristics of fast convergence rate of the cloud mutation model. And also, both the population size and the redundancy can be reduced. Then, the clonal selection algorithm is used to keep the diversity of the population effectively. Simulation results of the sliding mode control of discrete chaotic systems show that the improved PSO algorithm obviously improves the response speed, overshoot, and so on.
Chaotic system can be divided into integer order chaotic system and fractional chaotic system. Fractional chaotic system is a very complex nonlinear system [
Before introducing the sliding mode control of fractional order chaotic system, we mainly studied the control and synchronization of the fractional chaotic systems. At present, the control and synchronization of the fractional chaotic system are divided into two categories. One is the classical system stability theory and control method based on traditional integer order system. On the basis of the theory and control method, the stability analysis of the fractional order is carried out to design the integer order controller [ Feedback control based on pole placement of the fractional order system: Matignon [ The control of Gronwall inequality based on the fractional order: the Gronwall inequality is extensively applied to the integer order differential equation, and the solution of the system is expressed analytically. The Gronwall inequality is taken to analyze the system solution to obtain the design of the controller, which is a common control design method for integer order systems. Lazarević and Spasić [ Fractional order PID control: in the classical control theory and applications, PID controller has been developed very well and is widely applied to the actual system. In accordance with the PID control, the integral controller and differential controller of the PID controller can be partially extended to the fractional order; that is, the fractional integral and the fractional differential are introduced into the control design. Vasundhara Devi et al. [ Feedback control based on linear matrix inequality: Sabatier and others [ Sliding mode control of the fractional order system: the application of the integer order sliding mode to the fractional systems has been proved to be effective. The integral sliding mode surface and approach law are designed, respectively. And the effectiveness of the sliding surface is proved. The Lyapunov function is designed to analyze the stability of the fractional order system, and then, the sliding mode control law is obtained [
In this paper, a PSO multiobjective optimization algorithm is introduced. Since the algorithm is easy to fall into the local optimization, a sliding mode control of discrete chaotic system based on multimodal function series coupling is proposed combining the characteristics of multimodal function optimization problems. Simulations show the effectiveness of this method.
In many engineering optimization, such as complex system parameters and structural optimization, neural network weights, and structural optimization, we not only need to find the global optimal solution in the feasible region but also need to search several global optimal solutions and other valued local optimal solutions. Thus, multichoices and multi-information can be provided for decision makers. The above can be classified as multimodal function optimization problem or multipeak function optimization (MFO).
The multimodal optimization problem, as shown in (
In (
For the MFO problem of the objective function minimum value solution, the fitness function is defined as
In (
For the MFO problem of the function maximum value solution, the limit construction method is used to solve the minimization problem. The fitness function is defined as
In (
From (
Based on PSO algorithm and Powell method, this paper presents PP algorithm based on series coupling. The PP algorithm has the maximum repetition search
In the case of the particle swarm with the
In (
In the multimodal optimization, in order to make each local optimal value become the “only way” of some particles in PSO algorithm, this paper improves the standard PSO and obtains the improved PSO. Ideally, the self-cognition of each individual in the particle swarm represents each local optimization of the multipeak optimization. Thus, let Consider that the random function The linear decreasing strategy of inertia weight is simple and intuitive and has better searching ability, which makes the algorithm balance between the global search and the local search. Therefore,
In (
Based on the above improvements, (
The position change is performed according to (
The nonlinear Powell direct search method is a nonlinear direct local search method for solving unconstrained optimization problems without using derivatives. It is considered to be a relatively effective method in direct search. Powell search method is known as the direction acceleration method as well. Using this method, the calculation of the derivative is not required and only the function value is used to carry out the one-dimensional search from one point towards two directions. Then, the minimal value can be obtained. Nonlinearity means that the search direction of the initial point is not fixed, the acceleration direction will change with the change of the initial point position, and the search is not along a straight line.
The steps of the Powell search method are as follows:
The particle of the last search generation of PSO is taken as the initial point
According to (
In (
Let
According to (
When (
When (
The extreme points found by the PP algorithm are compared with all the extreme points in the extreme point database
The same peak judgement operator uses the peak exploration method to perform the same peak judgement towards the global extreme points. In this paper, the peak exploration method is applied in the same peak judgement of all the global and local extreme points. The method is as follows.
Set
In (
Let the numbers of particles
Comparisons of PP algorithm with other evolutionary algorithms.
Method | Number |
|
|
Time | |
---|---|---|---|---|---|
|
| ||||
ES | 1 | −2.9135 | 3.2658 | 648.9996 | 11.2 |
2 | 3.2584 | −1.5894 | 648.9996 | ||
3 | −3.4584 | −3.5814 | 648.9994 | ||
4 | 2.8595 | 2.1247 | 648.9991 | ||
|
|||||
PSO | 1 | −2.9251 | 3.2671 | 648.9997 | — |
2 | 3.2597 | −1.5902 | 648.9998 | ||
3 | 3.4595 | −3.5877 | 648.9998 | ||
4 | 2.8608 | 2.1211 | 648.9995 | ||
|
|||||
PP | 1 | −2.9295 | 3.2613 | 650 | 2.5 |
2 | 3.2657 | −1.5945 | 650 | ||
3 | 3.4624 | −3.5898 | 650 | ||
4 | 2.8695 | 2.1202 | 650 |
In Table
From Table
For function
Then, the global convergence of PP algorithm is analyzed.
Set
The change of the displacement of the two generations before and after the PP algorithm is
Due to the mutation operation of the basic normal cloud generator with normal distribution, the displacement of particle
Therefore, we can conclude that Theorem
In order to solve the problems of the multimodal function optimization, based on the above improved methods, this paper utilizes the characteristics of fast convergent speed of the cloud mutation model to compensate the shortcomings of fast training time of the PP algorithm. Meanwhile, it can reduce the population size and redundancy. And the clonal selection algorithm can effectively maintain the diversity of population. Hence, a particle swarm optimization algorithm based on cloud mutation clonal selection is proposed (WCPP).
The cloud model has the characteristics of randomness and stability tendency. The randomness can keep the individual diversity so as to search for more local extreme points, and the stable tendency can protect the better individual to perform adaptive localization of the global optimal values. Thus, in order to improve the accuracy of the PP algorithm and expand the search range to find other extreme points, a cloud mutation operator based on cloud model is introduced on the basis of the PP algorithm to update the particles.
Thresholds
In (
The cloud mutation condition can be defined as the degradation of a particle of
One-dimensional normal cloud operator can be extended to A normal random number
“Dimensional catastrophe problem” exists in dimension function. As the dimension increases, it is difficult to find the optimal solution in each dimension through the general evolutionary algorithms. The abnormal values on individual dimension lead to poor quality of the final solution or an inability to find the optimal solution to the optimization problem. Thus, in order to prevent the particles from falling into the local extreme points and failing to find the global optimal solution in high-dimensional multimodal function, the search particles are reinitialized with a certain probability
After the initialization, particles are updated according to (
In order to ensure the diversity of the population, optimization is performed based on the clonal selection algorithm. The steps are divided into clonal amplification, adaptive wavelet mutation, and immune selection.
In (
The antibodies in the antibody population were sorted in ascending order according to the affinity value before the clonal amplification. The clonal amplification operator for the sorted
In (
When
Morlet wavelet function.
With the progress of
In (
In (
In (
In (
From (
Select
Comparing the results of PP algorithm with WCPP algorithm.
Method | Number |
|
|
Time |
---|---|---|---|---|
PP | 1 | 3.5216 | 2.7526 | 91.5 |
2 | 3.5259 | 2.7568 | ||
3 | 3.5951 | 2.8692 | ||
4 | 3.6059 | 2.8925 | ||
|
||||
WCPP | 1 | 3.4253 | 2.2682 | 50.8 |
2 | 3.4158 | 2.2481 | ||
3 | 3.4682 | 2.2543 | ||
4 | 3.5060 | 2.4988 |
In Table
The mean value
Suppose a control system as
In ( the sliding mode exists: that is, ( the accessibility condition can be satisfied, and the movement points outside the switching surface will reach the switching surface with a limited time; the stability of the sliding mode movement is ensured; the dynamic quality requirement of the control system is satisfied.
If the first three basic problems can be satisfied, we can call the control as sliding mode control.
Henon fractional chaotic system is taken as an example; the sliding mode control model of fractional chaotic system is constructed. The Henon chaotic system with control is shown as
Define the switching function as
When the fractional order chaotic system enters the ideal sliding mode,
That is,
From the above, we can obtain the sliding mode control equivalent control part of the fractional order chaotic system
In (
Initialize the sliding mode controller, other parameters, and a group of particles with size
The evaluation function value of each particle is obtained by control of the discrete chaotic system according to the position of each particle and the optimal position of individual particle
If the repeated search algebra
The particles in the last search generation of the WCPP algorithm are cloned and amplified.
Adaptive wavelet mutation operator performance is carried out.
Immune selection performance is carried out.
And,
update particles
The searched optimal fitness particles are assigned to the initialized particles and turn to Step
In order to improve the fastness of the sliding mode control method for the discrete chaotic systems based on the WCPP algorithm, the values of the algorithm parameters
Figure
Sliding mode control of discrete chaotic system based on WCPP algorithm.
A discrete chaotic system is taken as the simulation object; the model of the controlled object is shown as
For the above discrete chaotic system, simulations of step input signal
Simulation results of discrete chaotic systems.
From Figure
For the above discrete chaotic system, the simulations of the square wave input signal
Square wave simulation of discrete chaotic system.
Another discrete chaotic system is also taken as the simulation object; the model of the controlled object is shown as
For the above discrete chaotic system, simulations of step input signal
Simulation results of discrete chaotic systems.
From Figure
For the above discrete chaotic system, the simulations of the square wave input signal
Square wave simulation of discrete chaotic system.
It can be seen from Figures
Chaos and chaos systems are widely used in physics, engineering, biology, finance, and so forth. The modeling of complex systems based on the theory of calculus can more accurately reflect the dynamic characteristics of the systems. Discrete chaotic systems can be widely used in the field of chaotic secure communications. Thus, it is of great theoretical and applicable value in the applications of control and synchronization of the chaotic dynamic systems. In order to overcome the shortcoming of the standard PSO algorithm in multimodal function optimization problems, a sliding mode control model of discrete chaotic systems based on the coupled multimode functions is proposed. The simulation results show that the improved algorithm proposed in this paper has significantly improved the response speed and overshoot performance compared with the standard PSO algorithm.
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 51675490).