Modelling of flexible beam based on ant colony optimization and cuckoo search algorithms

3.1. Ant colony optimization

In 1992, Marco Dorigo introduced a new SIA, which is ACO to solve computational problems based on travelling salesman problem [13]. This new algorithm was stimulated from the characteristics of an ant to discover the shortest distance to food source. When ants are looking for food, they automatically drop a chemical substance called pheromone on their pathway. The pheromone can be detected by other ants and will lead them to the food source. Therefore, when more ant passes on their trail, it becomes more attractive it will be followed by other ants.

Fig. 1The experimental setup for collecting input and output data [21]

Nevertheless, the basic ACO algorithm was developed using the discrete domains of the variables but due to that, it gives problems when it is related to variables laying in continuous domain. Thus, to fix the problems, ACO was extended in continuous domains. The principle of continuous ACO algorithm is still based on ACO metaheuristic without making changes of the main structure. So, the basic concept of ACO worksby advocating the construction of a solution based on the attractive pheromone solution adapted from the selection of component solutions. The discrete probability distribution was changed to continuous probability density function (PDF) in order to achieve the objective. Then, ACO employed Gaussian kernel which is a weighted summation of some Gaussian functions as PDF. Eq. (1) denote the formula of Gaussian kernel, $G^i$ [13]:

By derivation the $l$th member of the $k$ population size, the $l$th Gaussian function can be obtained. In each Gaussian function, it involves weights ($\omega$), means ($\mu$) and standard deviations ($\sigma$). After that, by sampling the $G^i$, a new $x_i$ solution is constructed. The ant is chosen probabilistically from a solution to achieve, $T$ which is proportional to weight, $\omega$ to develop a new solution using Eq. (2) [13]:

where $\omega$ is a weighted value of the solution with ranking $l$, 1.0 for mean and $qk$ is represented as standard deviation. The locality of search process, $q$ is one of the algorithm parameters which implies the weight value in solution archive. Therefore, $k$ is set as constant value, but a small value of $q$ means that the probability is biased to the best solution found whereas bigger value of $q$ gives uniform probability among solution in the solution archive. The probability of choosing the $l$th Gaussian function is given by Eq. (3) [13]:

A new random value is generated from the chosen PDF for each dimension solution by sampling Gaussian function using the equation. In this step, mean, $\mu$ and standard deviation, $\sigma$ from the PDF function are needed. The variables in each dimension of chosen solution are set as in Eq. (4) [13]:

where and the value of converge speed used $\xi >$ 0 is same for all dimensions in algorithm parameter which is used to adjust the size of standard deviation of Gaussian function. Standard deviation is described as the radius of search space for the sampling process. If the size of radius is bigger than 1.0, it will expand the search space. Meanwhile, when the size of radius is smaller less than 1.0, it will shrink the search space. The pheromone model in ACO is completed by ranking the solution archive, $T$. For each iteration, a new set of solutions is added to $T$ and ranked based on the fitness. In ACO, $T$ will always keep maintaining the number of populations, $k$ so it will be removed the worst set of solutions from it. This action related to the pheromone update in discrete ACO. Finally, the purpose of this is to make the search process bias towards the best solutions found during the search.

3.2. Cuckoo search algorithm

In 2009, Yang and Deb introduced another SIA namely CS. This algorithm was inspired by the behavior of cuckoo birds in placing their eggs in other birds’ species nest for survival purposes. Normally, parasitic cuckoo always aims for a nest where other bird species have just laid their own eggs. Generally, the cuckoo eggs will hatch before the host nest’s eggs, hence increasing the probability of the cuckoo chicks to receive food from the host bird. In addition, it is an advantage for the cuckoo chicks to have access to the feed since it can imitate the voice of the host bird’s nest.

The strategy from the breeding behavior of cuckoo can be used in solving the optimization problem. The solution is depicted by each egg laid in the host nest. In the meantime, a new solution is represented by a cuckoo egg. The purpose of the new solution is to establish a bigger possible solution by substituting the worse solution in the existing nest. In this study, each nest will consist of one egg since we only required a single objective function which based on MSE.

For the appropriate implementation, the cuckoo’s behavior can be idealized by considering three rules as follows [22]:

– At a time, each cuckoo lay and throws one egg at random host nest.

– The best nests that consist of high-quality eggs will be transferred to the next generation.

–The available number of host nest is fixed, and a host bird has a probability $pa\in$[0, 1] to recognize an alien egg. Due to this possibility, the host bird has 2 option which is either eliminates the egg or leave the nest and creates a new one in a new spot.

A set of random values in the lower and upper bound is assigned to each decision variable for the initial position of the nests. After that, the fitness is evaluated using objective function. Eq. (5) is used to calculate the initial position of each nest, $n$ [22]:

where $X_{ij}$ is several nests which represent the solution within the lower and upper limits. In addition, the range between 0 and 1 is assigned to the equation and denoted as $rand$. The next stage is generation of new cuckoos which represent the solution of the optimized problem. The number of cuckoos is the same as the number of nests since each nest represents only one solution. In this study, the exploration of search space is carried out by using two techniques which are known as Levy flight and random walk. An effective exploration in generating a new solution can be obtained using Levy flight because it has a probability distribution in random step lengths. It comprises a series of straight-line flight followed by an abrupt 90° turn. The Levy flights in CSA are carried out using Eq. (6) [22]:

where $x_i^t$ is an individual current location, $\alpha$ is the step size variable; $r$ is a random number resulting from a normal distribution and $x_{best}^t$ is the current best nest; and $S$ is random walk using Levy flights. A Mantegna algorithm is used to calculate the step length $S$ as shown in Eq. (7) [22]:

where $\beta$ is a parameter that has a range of 1 to 2 and for this study, it is considered as 1.5. The value of $u$ and $v$ can be gained from normal distribution as in Eq. (8) [22]:

The alien eggs discovery is performed for each component of each solution in terms of probability matrix as Eq. (9) [22]:

where $pa$ is the discovering probability. Existing eggs are replaced considering their quality by the newly generated ones from their current positions through random walks with step size using Eq. (10) [22]:

where $randperm1$ and $randperm2$ are random permutation functions used for different rows permutation applied on nests matrix and $P$ is the probability matrix.

4.1. Modelling of flexible beam using ant colony optimization (ACO) algorithm

The best model of ACO was obtained by heuristic method, which is by trial and error. In this algorithm, there were five parameters adjusted heuristically by substituting different values of population size, number of ants, intensification factor, number of iteration and model order. However, the two (2) parameters of ACO algorithm that were set as constant were the boundary [0.001, 0.001] and deviation distance factor [1]. Firstly, the tuning process was initiated by adjusting one of the parameters at one time and the other parameters was set as a constant value. Next, the other parameters were tuned until the best model of ACO algorithm was acquired. The best model of ACO algorithm was achieved when it satisfies the three validations which are lowest MSE value, has excellent correlation test and has high stability in pole zero diagram.

Table 1 represents the result of the best model parameters obtained from the ACO modelling method. In this study, the main criteria required for the best model it was that it must achieve the lowest MSE value. Based on the tabulated results, it was recognized that model order 3 achieved the lowest MSE value which is 1.0728×10^-8. Moreover, it also obtained 95 % confidence level of correlation test and has achieved good stability system in pole zero diagram. The finest model of ACO algorithm acquired can be classified as the simplest model based on the least number of parameters, which is 6.

According to Fig. 2 and 3, the developed model was capable to follow an actual measured output. Both graphs showed that the predicted output which overlapped the actual output can be declared as a good model for this system. Fig. 4 illustrates the errors of actual and predicted outputs of the system using ACO algorithm modelling. Finally, Fig. 5 and 6 shows the graph of both correlation test and stability of pole-zero diagram. The discrete transfer function obtained from the best model of ACO modelling was presented in Eq. (11):

Fig. 2Time domain for actual and predicted output of the system using ACO modelling

Fig. 3Frequency domain for actual and predicted output of the system using ACO modelling

Fig. 4Errors of actual and predicted outputs of the system for ACO modelling

Fig. 5Correlation test for ACO: a) auto correlation, b) cross correlation

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b)

Fig. 6Pole-zero diagram using ACO modelling

4.2. Modelling of flexible beam using Cuckoo search algorithm (CSA)

After that, there were five (5) parameters of CSA that were tuned which include the number of nests, the switching probability, lower and upper boundary, model order, and iteration. The initial tuning was focused on the CSA parameters and was followed with model order and finally the iteration. For number of nests, the values were varied from 16 to 40 while other parameters were fixed to a known value. Next, the value of switching probabilities was tuned from 0.01 until 0.5 while the number of upper and lower boundaries were tuned from [–1, 1] until [–8, 8]. In addition, the number of model order was tuned using RLS, which was 2 to 10. Finally, the iteration was tuned from 1500 to 5500 with an increment of 1000. The set of variables that has been tuned based on the heuristic method in obtaining the optimum model for a flexible beam structure via CSA was finalized and elucidated in Table 2. These parameters obtained the MSE, correlation test within a 95 % confidence level, and good stability in the pole-zero diagram.

The best CSA model managed to imitate the real system because the actual and predicted outputs overlapped with each other. This can be observed in Fig. 7 and 8 which shows the vibration experimental and simulation outputs for the system in time and frequency domains, respectively. In Fig. 8, the first mode of vibration is 2.279 dB and the error between the experimental and estimated output using CSA is highlighted in Fig. 9. The pole-zero diagram and correlation test for CSA modelling are shown in Fig. 10 and 11, respectively.

Fig. 7Time domain representation of actual and CSA outputs of the flexible beam

Fig. 8Frequency domain representation of actual and CSA outputs of the flexible beam

Fig. 9Errors between experimental and prediction outputs of the system for CSA modelling

Fig. 10Pole-zero diagram stability for CSA modelling

All three poles obtained from model order of 3 can be seen inside the unit circle which means that the system is stable. For the correlation test, the model was seen to be unbiased because the confidence level is between the 95 % interval. The discrete transfer function described in Eq. (12) represents the best model of flexible beam structure obtained using CSA modelling:

Fig. 11CSA model correlation test: a) auto-correlation, b) cross-correlation

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b)

4.3. Comparative performance model of ACO and CSA

The performance of developed model in ACO and CSA had been compared and discussed based on the 3 criteria’s which are MSE, correlation test and pole-zero diagram stability. The validation methods were chosen as it is capable to determine the accuracy of the system in many simulation environment [23]. Table 3 is the summary of the performance of modelling flexible beam using ACO and CSA algorithms. According to Table 3, the lowest MSE value for the ACO algorithm obtained was 1.0728×10^-8 while for the CSA, it was 6.1547×10^-9. Based on the MSE value, CSA is a more reliable algorithm due to the lower MSE value compared to ACO algorithm. Then, developed algorithms were validated using correlation test and pole-zero diagram to select the finest algorithm.

After that, in terms of the good correlation test, the result acquired must achieve at least 95 % of confidence level. As we know, it is impossible for real applications to achieve 100 % confidence level. Based on Table 3, it shows that the result of ACO and CSA models are unbiased because both systems are within the 95 % confidence level interval. Nevertheless, for the CSA model, it still proved to be the best model so far when it compares to ACO model because it has the lowest MSE value.

Other than that, the stability of the pole zero diagram is an indicator to determine the best model for the flexible beam. Therefore, the stability of the model can be determined based on the position of poles in the pole zero diagram. So, if all the poles are inside the circle, it can be considered as the most stable model. From Table 3, ACO and CSA models show that both algorithms have stable systems. These results were obtained from the observation that all the poles managed to plot inside the circle area. So, this study proves that SIA is more reliable compared to conventional algorithm.

Despite ACO and CSA models having met the requirements as the best model to represent the behavior of flexible beam structure, CSA model is more reliable due to the lower MSE. Hence, it can be summarized that CSA model is a suitable algorithm to represent the actual flexible beam structure. CSA algorithm proved to be the best model as it had the lowest MSE value, good correlation test, high stability, and the simplest model. Hence, the CSA’s transfer function was used for the development of the controller to suppress undesired vibration on flexible beam system.