A simple and intelligent online parameter identification of nonlinear chaotic systems

Abstract

Identification of nonlinear systems is one of the important problems in engineering. Chaotic systems are among those nonlinear systems that have been in the center of attention of many researchers because of their complex and unpredictable behaviors. This paper proposes an algorithm for optimally estimating the parameters of system by minimizing the mean of squared errors (MSE) index. In this paper, Teaching Learning Based Optimization (TLBO) algorithm is used for solving both offline and online parameter estimation problems for chaotic systems. The validity of this algorithm in terms of convergence speed and parameter accuracy in comparison to other popular optimization algorithms such as Standard PSO (PSO), Differential Evolution (DE), Adaptive Particle Swarm Optimization (APSO) and Genetic Algorithm (GA) is shown through an illustrative example for the modeling of chaotic systems. Furthermore, in order to demonstrate the feasibility of this algorithm, it is applied to the problem of parameter identification of a well-known nonlinear Lorenz chaotic system. According to simulation results, the proposed algorithm is a very suitable algorithm for online parameter identification for a class of nonlinear chaotic systems.

Keywords

Chaotic systems hybrid heuristic algorithm identification Particle Swarm Optimization (PSO)Teaching Learning Based Optimization (TLBO)Adaptive Particle Swarm Optimization (APSO)Differential Evolution (DE)Genetic Algorithm (GA)

1 Introduction

Chaotic systems are well known for their irregular and unpredictable behavior. These systems are complex, nonlinear, and extremely sensitive to initial conditions. Dynamic chaos is an interesting nonlinear effect that has been the subject of an intensive research in recent years. Chaotic phenomena are observed in many fields of science and technology such as biological systems, chemical systems electronic circuits, and power converters [1]. So far, various approaches have been proposed for synchronizing and controlling the chaotic systems [2]. However, if the parameters of chaotic systems are unknown, most of these techniques will be invalid. This is the reason that parameter identification for chaotic systems has been considered as an important issue in the last decade. Since chaotic systems are very complex, it may be difficult to determine their parameters.

In [3], an effort is made to estimate the parameters of a given dynamical model from scalar time series by adapting a computer model until it is synchronized with the given data. In [4], estimation of an unknown parameter was carried out by using two-valued symbolic sequences constructed by iterations of quadratic map when its initial value was known. In [5], the authors have introduced a linear feedback to synchronize system variables also; they have proposed an adaptive control method for estimating one of the parameters of the transmitter for chaotic signal communication in order to guarantee the security. In [6], the authors have introduced an adaptive control-based synchronization approach which has later been used in the same paper for parameter identification for a modified chaotic van der Pol-Duffing oscillator. Also, for the parameter identification problem of chaotic systems, an evolutionary programming approach has been proposed in [7].

So far, various types of classical methods have been developed to solve these problems. Among these techniques, meta-heuristic-based methods such as the particle swarm optimization (PSO) algorithm, genetic algorithm (GA), and the differential evolution algorithm (DE) may be the most popular ones that formulate the problem as a multi-dimensional optimization problem [8].

Recently, a novel optimization technique known as ‘Teaching Learning-Based Optimization’ (TLBO) is proposed by Rao and Patel [9]. Rao and Patel clearly explained that TLBO is an algorithm which doesn’t need specific parameter [10]. In [11], five different constrained benchmark functions with different characteristics, four different benchmark mechanical design problems, and six mechanical design optimization problems with real world applications have been used to evaluate the effectiveness TLBO method. The TLBO algorithm is easy to understand and implement; it requires few parameters and has been empirically shown to perform well on many optimization problems [12 –14]. Its application has been extended to function, engineering, and multi-objective optimization problems [15, 16]. A brief survey of the literature shows other recent developments and applications of the TLBO.

So far in literature, the TLBO approach has been used for optimizing the parameters of various advanced machining processes such as electrochemical discharge machining (ECDM) process and electrochemical machining (ECM) process so that all the objectives of these processes could be satisfied by the optimized parameters [11]. This technique has also been used for optimizing field specific process functions in different fields e.g. it has been used to optimize the parameters of fuzzy and PID controllers [17, 18]. Also, several modifications have been made to original TLBO technique in order to develop a multi-objective TLBO algorithm which has been used for solving the dynamic economic emission dispatch problem [19].

In solving an optimization problem, the convergence speed of any evolutionary algorithms is often considered to be more important than the quality of solutions. Generally, the TLBO technique produces better results in comparison with the other evolutionary computation (EC) approaches such as Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Differential Evolution (DE), and Artificial Bee Colony (ABC) [19]. However, the convergence time is a proper of greater importance in real world real time applications. Therefore, In order to make TLBO suitable for such applications, this research focuses on achieving a higher convergence speed without decreasing the quality of results.

The aforementioned studies motivated the authors of this paper to present a TLBO-based evolutionary computation algorithm for both offline and online parameter identification of chaotic systems. As far as we know, this is the first attempt that is made so far to apply TLBO for online parameters identification of chaotic systems. The proposed algorithm introduces a Teaching Learning Based Optimization to achieve better solution accuracy and higher convergence speed with minimum incremental computational burden; properties which make it more appropriate for online identification. In order to show the feasibility of this algorithm, it is used for identifying the parameters of Lorenz chaotic system. And in the case of offline identification, the performance of the proposed TLBO in terms of convergence rate and parameter accuracy is compared with those of standard particle swarm optimization (PSO), Differential Evolution (DE), Adaptive Particle Swarm Optimization (APSO) and genetic algorithm (GA). Simulation results show that the proposed algorithm outperforms GA, PSO, DE and APSO.

2 Problem statement

The structure identification problem becomes a difficult problem unless we have a prior knowledge about the real system otherwise, the structure must be selected by trial and error. Fortunately, there is a great deal of knowledge about the structures of most industrial processes and engineering systems and we are often able to derive a particular class of models are capable of accurately describing the real system. Therefore, the system identification problem is often reduced to a parameter estimation problem [20 –23]. In this paper, the following n-dimensional nonlinear system is considered to explore the parameter estimation problem: $\dot{X} = F (X, X_{0}, θ)$ (1)

In Equation above, X = [x₁, x₂, …, x_n] ^T ∈ Rⁿ represents the state vectors, X₀ is the initial states, θ = [θ₁, θ₂, …, θ_n] ^T ∈ R^m shows the unknown parameters vector and F : Rⁿ × R^m → Rⁿ is a given nonlinear vector function. Also, an estimated model is defined as follows to estimate the unknown parameters in (1):

\dot{\hat{X}} = F (\hat{X}, X_{0}, \hat{θ})

(2)

where, $\hat{X} = [{\hat{x}}_{1}, {\hat{x}}_{2}, \dots, {\hat{x}}_{n}] \in R^{n}$ is the estimated state vector and $\hat{θ} = [{\hat{θ}}_{1}, {\hat{θ}}_{2}, \dots, {\hat{θ}}_{n}]^{T} \in R^{m}$ denotes the estimated parameter vector. An objective function needs to be defined before the initialization of heuristic algorithms because these algorithms only depend on the objective function to guide their search process. This study considers the mean squared errors (MSEs) between estimated and real responses for a number of given samples as the fitness of estimated model parameters. Therefore, the following objective function is selected:

MSE = \frac{1}{N} \sum_{k = 1}^{N} e^{2} = \frac{1}{N} \sum_{k = 1}^{N} [X (k) - \hat{X} (k)]^{2}

As shown in above equation, N shows the sampling number, and X (k) and $\hat{X} (k)$ represent the real and estimated values at time k, respectively. The goal of this research is to use the proposed TLBO algorithm for minimizing the MSE value so that the parameters of the actual nonlinear system are estimated accurately. A typical block diagram of nonlinear system parameter estimation is shown in Fig. 1. It could be seen in this figure that the initial state is fed to both the real system and the estimated model [23]. Then, the outputs of both the real system and the estimated model are used as the inputs of the optimization algorithm in which the objective function i.e. MSE will be calculated.

3 Teaching learning based optimization

Teaching-learning based optimization algorithm (TLBO) is proposed by Rao et al. [9]. This is based on the effect of influence of a teacher on the output of learners in a class. The algorithm mimics the teaching-learning ability of teacher and learners in a classroom.

Moreover, learners also learn from the interaction among themselves which also helps in improving their results. By using the mentioned process, TLBO algorithm is introduced for solving various optimization problems. It has two basic modes of learning, through teacher (known as teacher phase) and interacting with the other learners (known as learner phase). In training stage, initially the best solution is considered as a highly learned person between population members (population is a random set of solutions that is produced in search space which moves to the best solution for objective function through the optimization steps) called teacher until end of this optimization step. Then the mean of position of total population is calculated as a criterion of achieved information from all population members. Now, all members of population should move toward teacher’s training who considers the mean of class as the training basis. In order to achieve this goal, updating of population is done by the belowequation:

x_{i}^{k + 1} = (x_{i}^{k} + rand (Teacher - TF \times Mean))

(3)

Where TF is the teaching factor

TF = (1 + rand)

(4)

And rand is a random number in the range (0, 1).

After position updating affected by teacher’s training, cost function is calculated for the new solutions. If this solution is better than the previous one, the new solution is chosen as the new position of that member, otherwise the new generated solution would be omitted. This process iterates for all members of population at the teacher’s training stage. After this step, it’s the time to train the learner. In this stage, each learner (j) update his position by means of interacting with another learner who is selected randomly from the members of population, so he increases his knowledge using that learner’s information. Position updating in this step is done as below:

x_{j}^{k + 1} = (x_{i}^{k} + rand (x_{j}^{k} - x_{i}^{k}))

(5)

If the cost function of new position is better than the previous one, the new solution is chosen as the new position of that learner, otherwise it would be neglected. It should be mentioned that if a better solution rather than the teacher was found in position updating, teacher should be replaced with it. With implementation oftwo-step teacher and student learning to all members of the population, a step of repetitive steps of optimization algorithm in the process ends.

Then the convergence condition of problem or the condition of closing of the optimization is investigated. In this case, if this condition has been satisfied, the running of algorithm of program ends and the last opportunity for the teacher is fixed as the final answer. Flowchart of the algorithm can be seen in Fig. 2.

4 Particle swarm optimization

Unlike population-based evolutionary algorithms, PSO is inspired by the social behavior of swarms of fishes or flocks of birds to find a good food place, and a velocity is associated to each candidate solution [24 –31]. The candidate solutions that are called particles in this algorithm fly through the search space. First, a population with the same size as particles is generated. Then, the velocity of each particle is constantly updated based on the corresponding particle’s experience and the particle’s companions’ experiences [21].

By doing this, it is expected that the particles will move towards areas of better solution. Then, using the objective function of optimization problem, the fitness of every particle is evaluated. The velocity of each particle will be calculated at each iteration by using the following Equation:

\begin{matrix} v_{i}^{t + 1} = {wv}_{i}^{t} + c_{1} r_{1} ({pbest}_{i}^{t} - x_{i}^{t}) \\ + c_{2} r_{2} ({gbest}^{t} - x_{i}^{t}) \end{matrix}

(6)

where in the equation above, $x_{i}^{t}$ denotes the position of the particle i in tth iteration, ${pbest}_{i}^{t}$ represents the best previous position of particle i (memorized by every particle), gbest^t is the best previous position obtained among all the particles in tth iteration (memorized in a common repository), w shows the inertia weight, parameters c₁ and c₂ are the coefficients of acceleration and are called cognitive and social parameters, respectively. Finally, r₁ and r₂ represent two random numbers within the range [0, 1]. Once the velocity is calculated, the new position of every particle can be calculated as follows:

x_{i}^{t + 1} = x_{i}^{t} + v_{i}^{t + 1}

(7)

The update equations above are repeatedly used by this algorithm till the pre-specified number of generations G is reached. Despite the fact that standard PSO has shown some great advances by providing high convergence speed in particular problems, it does have some drawbacks. Researchers have discovered that standard PSO offers a poor capability in searching for a fine particle since it does not have a mechanism to control its velocity [17].

4.1 Adaptive PSO (APSO)

The aforementioned reasons motivated the authors of this paper [21] to dynamically adapt the inertia weight of each particle by using a measure known as adjacency index (AI) that shows how close an individual fitness is to the real optimal solution. Using this measure, each particle becomes able to decide how to adjust its inertia weight value. The following Equation shows the velocity updating rules in the proposed APSO:

\begin{matrix} v_{i}^{t + 1} = w_{i}^{t} v_{i}^{t} + c_{1 i}^{t} r_{1} ({pbest}_{i}^{t} - x_{i}^{t}) \\ + c_{2}^{t} r_{2} ({gbest}^{t} - x_{i}^{t}) \end{matrix}

(8)

In order to calculate the inertia weight of ith particle in tth iteration which is denoted by $w_{i}^{t}$ in Equation (8), first the adjacency index (AI) is defined as follows:

{AI}_{i}^{t} = \frac{F ({pbest}_{i}^{1}) - F_{KN}}{F ({pbest}_{i}^{t}) - F_{KN}} - 1

(9)

where in the equation above, the fitness value of the best previous position of ith particle i.e. the best position found so far by the ith particle is denoted by $F ({pbest}_{i}^{t})$ , and the value of real optimal solution is denoted by F_KN. It could be concluded that AI is an index that varies with number of particles, and it is set based on the feedback taken from the best memories of particles. A small value of shows that the fitness of ith particle is located far away from the real optimal value and as a result, it requires an effective global exploration therefore, the value of inertia weight must be large. Conversely, a big value of AI shows that the ith particle is very close to the real optimum and therefore, it requires a strong local exploitation, and as a result the value of inertia weight must be small. Thus, in tth iteration, the inertia weight of each particle is calculated dynamically by means of the followingformula:

w_{i}^{t} = \frac{1}{1 + e^{- (α \times {AI}_{i}^{t})^{- 1}}}

(10)

In the above Equation above, α is positive constant within the range (0, 1]. Based on the definitions and assumptions above, it can be concluded that 0.5 ≤ w_i < 1. It is obvious that the inertia weight value of each particle in tth iteration depends on the value of α. The decreasing speed of inertia weight is controlled by the value of this parameter so that a lower value of parameter α, results in a higher decreasing rate of inertia weight. The value of parameter α is changed from 0.1 to 1 with a step size of 0.1, in order to see the effect of this parameter on the variation of inertia weight.

Based on Equations (9) and (10), in this algorithm, the particles face different fitnesses during the search process; consequently, they gain different values of AI and then different values inertia weight. When a particle’s fitness is far away real global optimal the AI of this particle will have a small value i.e. a low adjacency, and the value of inertia weight will also be large which result in strong global search capabilities and find the promising search areas. On the other hand, when the fitness of a particle gets close to the real global optimum, the AI value of this particle becomes big (i.e. a high adjacency) and the inertia weight will be set small, depending on the closeness of its best fitness to the optimum value, for facilitating finer local explorations and so accelerating the convergence.

5 Simulation and results

In this section, to prove its feasibility, the TLBO is used to solve the parameter identification problem of a chaotic system. Chaotic systems are deterministic nonlinear systems. One of the salient features of these systems is that they sensitively depend on initial conditions. Because of the complexity of chaotic systems, it might be difficult to determine the parameters in real world applications. Therefore, during the last decade, the problem of parameter identification for chaotic systems has been the subject of an intense research [21, 22]. Lorenz system is a famous example of this kind of systems. It was first introduced in 1961 for modeling the unpredictable behavior of the weather. It has been shown that various physical systems such as laser devices and disk dynamos, also several problems related to convection could be described by Lorenz dynamic equations given below:

\begin{matrix} \dot{x_{1}} = a (x_{2} - x_{1}) \\ \dot{x_{2}} = {bx}_{1} - x_{2} - x_{1} x_{3} \\ \dot{x_{3}} = x_{1} x_{2} + {cx}_{3} \end{matrix}

where in the Equation above, x₁, x₂ and x₃ represent the states of the system, and θ = [a, b, c] are the unknown parameters of the system that must be identified. Simulations are performed in two stages. In the first stage, in term of accuracy, the TLBO is compared with PSO, GA and APSO in offline parameter identification. Next in the second stage, the TLBO is applied to online parameter identification. In this study, [x₁ (0) , x₂ (0) , x₃ (0)] ^T = [1, 1, 1] ^T is considered as the initial conditions. At this stage, it should be reminded that the goal is to determine the parameters of the system by means of the aforementioned optimization algorithms i.e. GA, standard PSO, APSO, and TLBO, so that the value of objective function F given in Equation (3) is minimized, getting as close to zero as possible.

5.1 Scenario 1. Offline identification

In this scenario, the original parameters of Lorenz system are considered to be equal to θ = [a, b, c] = [10, 28, 8/3]. Also, the goal of this parameter identification is to determine θ as precisely as possible. In order to compare the TLBO algorithm with GA, standard PSO, APSO and DE under fair conditions, the same computational effort is used in all these algorithms. This means that, the maximum number of iterations and the population size of all these algorithms are set to 200 and 20, respectively (Since in each iteration of teaching and learning based optimization algorithm (TLBO) the objective function value is calculated twice, so the number of iterations of the algorithm to other algorithms studied half and sets as 100). Two different cases are simulated and each of the aforementioned algorithms is in implemented 50 times independently for both cases. Table 1 shows the results that obtained from scenario 1 which considers three-dimensional parameter identification where none of the parameters in Lorenz system are known and need to be identified. It can clearly be seen in Table 1 that even the best results obtained by PSO, GA, DE, and APSO are worse than the worst results obtained by TLBO. Furthermore, it can also be observed that all the values identified TLBO are very close to right values.

As it can be seen in Table 1, compared to PSO, DE, APSO, and TLBO, the results of GA are far away from the right values. These results reveal that the proposed TLBO is more robust and more effective than the other algorithms in identifying the parameters for Lorenz chaotic systems. Figures 3– 5 show the trajectories of parameters and objective function F for PSO, APSO, DE and TLBO. Also, Figs. 3– 5 show that the trajectories of all estimated parameters a, b and c converge to their actual values, respectively. As it is shown in these figures, all parameter a, b and c quickly converge to their right values, which shows that TLBO has he great efficiency in achieving global optimum, compared to APSO and PSO. It can be seen in Fig. 6 that the value of F falls very quickly to zero implying that TLBO can converge to the global optimum.

5.2 Senario 2. Online identification

In this scenario, the variations of system parameters are taken into account. As it was mentioned before in Section 5.1, in terms of accuracy and convergence speed, the proposed TLBO algorithm outperforms GA, PSO and DE in offline identification. This is the reason that TLBO is applied for online identification of Lorenz system parameters.

With the beginning of the identification process (by the start of the identification process), the algorithm according to the objective function is defined to enter the cycle repeats itself and the iterations continue until the answer was found less than the predetermined success (10^-6). To reduce the computational load of the processor system, the algorithm stops at each time step and then check out the error rate corresponding to the best answer. If the error rate is much more determined to succeed the algorithm detects changes in system parameters, and the definition of a new population found around the previous answer, again, is to start. To achieve this standard algorithm stop repeatedly to make changes in the system parameters of the algorithm check out the last error responses, always in a standby state observer, remain.

New initial population after any change in the parameters is produced in accordance with the following formula.

\begin{matrix} x_{i}^{t} = {Gbest}^{old} + 2 (rand - . 5) \frac{(X_{max} - X_{min})}{4} \\ \forall i = 1, \dots, N . \end{matrix}

This study has to be done in order to justify tracking the changes of system parameters by means of the proposed algorithm. Thus, an abrupt change is applied to the system parameters. The parameters values are selected in way that the chaotic behavior of Lorenz system could be guaranteed. The range of parameter values that causes the Lorenz system to show a chaotic behavior is approximately 0 < a < 20 , 0 < b < 50 and 0 < c < 5 [21]. Figures (7– 10) show the way in which the proposed algorithm tracks the variation of system parameters. In the beginning, the rated values of parameters are used and the proposed TLBO algorithm detects these parameters if they reach a threshold value of 10^-6. The TLBO algorithm runs further if any change occurs in the parameters. At this stage, two abrupt changes are applied to the parameters of the Lorenz system in order to show the effectiveness of the proposed algorithm in tracking the time-variant parameters. The time that some changes in system parameters are detected by the particle has been shown by dashed lines in Figs. 7– 9. At the first stage, at t = 50 seconds, a is varied simultaneously from 10 to 11 while the parameter b and c are changed from 28 to 27 and from 8/3 to 2, respectively. At the second stage, at t = 100 seconds, a, b and c are changed from 11 to 13, 27 to 30 and 2 to 3, respectively. Figures 7– 9 show that the proposed algorithm is capable of tracking any change in parameters. Furthermore, it can be observed that when simulation is stopped at the end of each stage, unchanged parameters i.e. b in the first stage, and a and c in the second stage reach their previous values. The results obtained by TLBO in online identification are illustrated in Table 2. These results show that TLBO has successfully identified the changes in parameters.

6 Conclusion

In this paper, online parameter identification for nonlinear systems was converted to a multi-dimensional optimization problem. Online estimation of systems is often a challenging problem especially in cases that a system with unknown and varying parameters is involved. In online implementation, there are some difficulties that mostly stem from the inevitable computational time for finding a solution. Therefore, a new optimization approach known as TLBO which is inspired by the nature was employed to solve this problem. in order to investigate the feasibility of the TLBO, it was used to identify the parameters of a Lorenz system, which is a chaotic nonlinear system Simulation results confirm that in terms of accuracy and convergence speed, the TLBO algorithm outperforms GA and some alternative PSO algorithms and DE without the facing premature convergence problem. Simulation results and comparisons made proved that the proposed algorithm is able to identify time-variant parameters successfully.

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