Spider-based FOPID controller design for temperature control in aluminium extrusion process

3.1.1 PID Controller

Basic structure of the PID controller is given by

where G c is the transfer function of the controller, K p is the proportional gain, K i is the integral gain, and K D is the derivative gain.

In the PID controller, the summation of the proportional error signal, the integral error signal and its derivative error signal gives the system output. Here, the error signal is the subtraction between the reference input temperature and the actual plant output temperature. Adding a controller leads to an increment in order and type of the system, which results in lowering steady-state error. Thus, the PID controller improves the stability of the system. For a linear time-invariant temperature plant, the PID controller equation is written as,

where e(t) = r(t) – y(t) and K = 0, 1, 2, …, n.

If k tends to ∞, the error is minimized to zero.

Controller parameters and minimization of the objective function are tuned by PSO and Spider algorithms.

3.1.2 Fuzzy PID Controller

The conventional PI and Derivative controller can be extended into a Fuzzy PID controller by making adjustments in the PID parameters by fuzzy control. The above equation can be rewritten into a fuzzy PID controller, and the equations are as follows:

Where K P ′ , K I ′ , and K D ′  – initial PID gain values, k _p, k _i, k _d – adjusted PID gain values.

A fuzzy set of error e, and rate of change of error de is utilized to define the PID parameters. The fuzzy membership function can be formed as negative big (NB), negative small (NS), zero (Z), positive small (PS), and positive big (PB). Based on nonlinear Gaussian distribution [15], the membership function is defined. Then, using these membership functions, suitable fuzzy rules of k _p, k _i, k _d are formed according to the experiences. The fuzzy rule of k _p is shown in Table 1.

According to fuzzy tables, 25 × 3 fuzzy rules are formed with the following statements.

If error (e) = A _i and rate of change of error (de) = B _j then k _p (k _i, k _d) = C _ij .

where A _i is the fuzzy sets of error; B _j is the rate of change of error; and C _ij is the PID parameters K _p, K _i, K _d.

With these Fuzzy rules, the fuzzy PID parameters K _p, K _i, K _D are obtained. For tuning, gain values of PID controller and minimization of error PSO and proposed Spider algorithms are used.

3.1.3 FO controller

In order to implement the fractional controllers in practice, the integer-order approximation of the fractional derivatives and integrators is used because the exact implementation of fractional-order operators in online applications is not possible due to their infinite memory characteristics [26]. This approximation may cause a difference between the behaviour of the implemented system and the expected behaviour of the closed-loop system. Therefore, the use of an appropriate approximation is necessary for the proper implementation of fractional-order operators. The main purpose of this study is to concentrate on providing a set of tuning rules for the implementable form of FOPID controller by using the algebraic formulation of ISE cost functions in terms of free parameters of the fractional-based controller. In doing so, first, the integer-order realization of the FOPID controller, which is referred to as an implementable form, is obtained.

Li et al. have expressed that the fractional calculus is globalization of integration and differentiation for a number that is not a whole number, a negative whole number, or zero-order fundamental operators [18].

where r ∈ R . But (r may also be a complex number) r is the order of the operation.

Based on Equations (2) and (3), the FO system has been defined as

FOPID structure for aluminium extrusion process is given by

where

e(t) = ISE = ∫ o t r ( t ) − y ( t ) .

Instead of three controller parameters in the PID controller k _p, k _i, k _d, five controller parameters are tuned in FOPID controller k _p, k _i, k _d, λ, μ where λ and μ are the integral and derivative orders. This value lies between 0 and 1. If the orders are equal to 1, then FOPID works on the conventional PID. Similarly, PSO and spider algorithms are used to tune these parameters and finally, and ISE gets minimized.

3.1.4 PSO Algorithm

PSO is an uncomplicated optimization algorithm, and it has been effectively applied to a number of applications in several fields of engineering and science such as operations research, mesh processing, data mining. A new method for optimal control of parameters in PSO based on fuzzy rules is presented. In other words, fuzzy linguistic variables and membership functions are employed to conduct the swarm towards the global optimum point. Several computational simulations are carried out to demonstrate the high performance and stability of this method. Simulation results reveal superior optimality and stability and lower computational cost of the new algorithm compared to the traditional meta-heuristics such as standard PSO, which justifies its advantages for PSOs. It is mainly applicable for continuous search spaces. In other words, to apply this algorithm in some state spaces, modifications to standard formulation are required. Thus, in the proposed system, the optimized controller gain and order parameters can be obtained from the search space.

The position and velocity equations are given by

where ω V i j ( t ) is the inertia term, c ₁ and c ₂ is the accelerating coefficients, r 1 c 1 ( P i j ( t ) − X i j ( t ) ) is the cognitive component, r 2 c 2 ( g j ( t ) − X i j ( t ) ) is the social component, and r 1 and r 2 ∼ U ( 0 , 1 ) (Table 2).

Conventional PID, Fuzzy PID, and FOPID controller parameters gain values and order values are tuned by the PSO algorithm. Therefore, the ISE can be minimized with the tuned values.

3.1.5 Spider-based controller algorithm

Spider algorithm is different from other algorithms like PSO, GA, and EA. Unlike all SI algorithms, the spider is self-organized, because solutions are processed based on their gender. Since the female and male spiders accomplish wide examination and extensive exploitation, the algorithm avoids premature convergence and incorrect exploration. All particles pursue their own best position and a common global best position in PSO. On the other hand, social spiders in algorithms chase positions through the knowledge of others’ current positions and those positions which it is not sure to be already visited by the population before. This feature improves its ability to solve optimization problems with many local optimums but makes its convergence slower. While all particles in PSO are conscious of the system information without any loss, the spiders communicating through vibrations form an all-purpose information scheme with information loss.

In the Spider algorithm, the spider is symbolized by widespread search space from a spider web. Among the total population, all candidate solution is represented by a spider. Based on the fitness solution, all individual spiders collect a weight from the fitness function. In a colony, two-non identical sets of evolutionary operators imitate the different mutual performances of the spider. Nonlinear global optimization problem having box constraint was assumed for spider algorithm. The optimization problem was represented by

Minimize

where f: R ^d → R is a nonlinear function

which is a minimized feasible space bounded by the lower ( l h ) and upper ( u h ) limits. Bound performs a strong guarantee; they never worsen the objective function. This structure is often exploited to obtain a bound on the objective function and proceed by optimizing this bound. Ideally, a bound is looked for as a valid type in each and every parameter space, easily optimized, and equal to the true objective function at one (or more) point(s) [27]. This approach results to be effective in identifying thresholds better than an exhaustive search. In general, when the number of levels of thresholding is increased, performance in terms of stability and accuracy deteriorates. But, SSO distinguishes itself by stability and accuracy (Figure 3).

Figure 3

Flowchart for spider algorithm.

The optimization problem is solved from S of N candidates. From X search space, spider position represents the solution. In this method, the population S is segregated into two search representatives, namely, Male (M) and Female (F). Let us assume N f is the number of females, and it is aimlessly selected in a value range of 65–95% of all-inclusive population S, while the others are assumed and considered as male individuals ( N m = S − N f ) . From the above condition, a set of female individual’s ( F s = { f s 1 , f s 2 , … f s Nf } ) groups is formed. Similarly, the male individual set M s is framed such as, ( M s = { m s 1 , m s 2 , … m s nm } ) , where

Each spider's weight we _i is calculated based on a fitness function, and that weight is planned as follows:

where fit i is the fitness of the ith spider posture, i ∈ 1,., N.

In SSO optimization, information swapping is the basic mechanism. From vibrations produced on the spider web, it is simulated. The vibration equation is modelled by

where we _j is the weight of the jth spider and d is the distance between two spiders.

Each and every spider “i” can be distinguished by three types of vibration, v i , n , v i , b and v i , f . v i , n is the closest spider vibration of n having a highest weight produced ( w n > w i ) . v i , f is the nearest female spider if “i” is a male spider. v i , j is the best and perfect spider among the population S.

A population of N spiders is steered at initial stage k = 0 to ( k = i t ) . The new position of female spiders f s i k + 1   is acquired by changing the current spider position f s i k . The adaption is arbitrarily maintained by the movement constructed in link to other spiders, and its vibration and probability factor p f through the entire search space:

where k is the iteration number; ∝ , δ , and rand δ are random numbers between (0, 1); and s b is the best spider in the communal web. s n is the nearest spider with a higher weight than f s i k .

Male spiders are categorized into two categories namely D (dominant) and ND (non-dominant) spiders. Generally, D is the set which is consolidated with the male spider fitness value set. Also, ND is a group which is constructed by the rest of the male spiders. Based on the following equation, the male spiders m s i k are steered by the following equation:

where s f is the nearest female spider to male individual i. ∝ , δ and rand are random numbers between (0, 1).

Figure 4 illustrates the complete evolutionary process in the form of a flowchart. In the suggested algorithm, dominant male m d and female are mating between individuals with a fixed range r. A new individual set s new is generated. The weight of an individual spider is described by the probability control of each and every spider on s new . The main element has an increased probability to control the new individual s new . If a brand new spider is formed, this is compared and contrasted with the balance population. If the latest spider has an improved fitness value compared to the poorest spider membership function, the weaker spider is restored with s new ; else, s new is removed.

Figure 4

Weight updating process for spider algorithm.

In the spider-based controller, the algorithm gets input from the sensors of the aluminium extrusion plant and feedback output from the aluminium extrusion plant. Then, the error output signal is found, which is a switching sequence akin to a web-building behaviour of a spider. In the spider’s web-building process, in the beginning, it forms a silky thread from its mouth. Akin to that, the controller receives input signals such as input temperature Y _i(t) from the thermal sensor output. Then, after forming silk, the spider thrust it in the air to cling to some fixed place. The place where it clings to is based upon the parameters such as the distance from the spider and the strength of the silk [28].

In the beginning, the spider has the temperature output (thread), it peruses the value of the temperature at each and every instant (explore the place to fix silk), and after that, it moves to the zero crossing, which is the most suitable place where it can fix the silk. And again, it moves to another location which forms the duty ratio and fixes the silk. It moves back to the past location by eating the old thread. This operation gives the temperature ON and OFF cyclic process. Using equations (20) and (21), the minimization of the objective function and also the controller parameters are obtained. The flowchart for the spider algorithm is shown in Figure 5.

Figure 5

Temperature control flowchart using spider algorithm.