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Abstract

The optimization algorithms for identification and control processes take more and more place in the industrial domain. However, determining a control law for fractional systems remains challenging for researchers today. This paper presents a simple method for using a fractional PID (FOPID) corrector to regulate a class of fractional systems that show a stable step response. The procedure is based on an algorithmic identification to assimilate the fractional system into a simple model. Optimization methods were used. Using techniques like Particle Swarm Optimization (PSO), Genetic Algorithm (GA), Artificial Bee Colony (ABC), and Ant Colony Optimization (ACO) to ascertain the parameters of the basic model, a first-order system with delay. Then, the FOPID controller will be calculated using Ziegler–Nichols methods and optimization algorithms to stabilize the real process. Better performance Overshoot and Settling time as well as minimal Energy control effort are benefits of the proposed design. To confirm that the fractional regulator is robust which is optimized by the best-chosen method (PSO), the parameters of the chosen process will be randomly varied. Through two numerical simulations, the efficiency of the suggested method has been confirmed.

Keywords

FO-PID controller stability of first delay system effort PSO GA robustness

Introduction

Fractional models have gained popularity in the past few decades to reflect a wide range of real-world issues.¹ Identification of thermal systems² and identification of lithium-ion batteries³ are accomplished by the application of fractional calculus models and optimization parameters.⁴ Other natural processes that fractional differential equations or fractional integrals can represent include heat conduction in a semi-infinite slab,⁵ fluid flow in a porous material, and the voltage-current connection in a semi-infinite transmission line. Conversely, it seems that the study of the stability of fractional systems is sensitive and requires a great deal of mathematical computation. Numerous scholars have utilized newly developed theorems.^6,7 Since we are talking here about fractional systems, it seems logical to try such a fractional controller as the fractional PID (FOPID).⁸ An expanded variant of the traditional integer order PID was proposed by Podlubny,⁹ the first iteration of non-integer order PID (FOPID or PIλDμ). Based on the fractional computation, the five parameters that constitute the fractional PID controller are ( $K_{p}$ ), ( $K_{I}$ ), integration order (λ), ( $K_{d}$ ) and derivative order (μ). When compared to the normal PID controller, both two parameters, λ, and μ, increase the fractional PIλDμ controller’s efficiency.^10,11 However, their design and implementation are more intricate, Thus, it takes greater attention to fine-tune these five characteristics. When developing and fine-tuning the original PID corrector for first-order time delay systems, automation engineers mostly employ the Ziegler and Nichols technique, originally introduced in 1942.

In this work, we introduce a novel technique of tuning parameters FOPID to stabilize a fractional system. First, we look for a first-order system with a delay equivalent to our fractional system. This identification phase must be very precise to have a model that is as close as possible to the system to be studied. This article presents a set of identifications with optimization algorithms,¹² such as the genetic algorithm, Artificial Bee Colony (ABC), and others. They give good performance, but particle swarms PSO algorithm¹³ is gaining more and more the trust of researchers. It is suitable for all business sectors. When optimizing the parameters using these techniques. The output error between the simplified version and the original system forms the foundation for the effective fitness function.¹⁴ Secondly, through the extension of the Ziegler–Nichols rules developed by Valerio and Costa,¹⁵ we determine the parameters of the FOPID regulator Poor performance is the primary drawback of this tuning method, especially for high-order systems (high overshoot and extended settling time). Therefore, this study about optimizing the FOPID corrector’s parameters for self-tuning in fractional system control piques our curiosity.

The major challenge in control engineering is reducing the effort required to stabilize a system while maintaining performance at an acceptable level. For this, we show in this study that our adjustment and identification method allows an interesting saving of effort.

The mathematical model is typically only a crude approximation of the actual physical system since it operates under preset theoretical parameters.¹⁶ Because of incomplete or erroneous measurements, component deterioration, and/or environmental changes, these parameters are frequently not precisely defined.¹⁷ Examining whether the closed-loop system stays stable in the presence of unclear and changing parameters is an intriguing question. Our objective is to design a model and a controller that are less sensitive to external disturbances. Reliable performance requires a system to be stable under a variety of conditions, but robustness is also demonstrated by the system’s capacity to maintain desired performance in the face of external disturbances.

This work’s remaining sections are arranged as follows: Fractional calculus and the derivation of fractional order dynamic systems are covered in Section “Fractional order dynamic systems and fractional calculus presentation.” Section “System identification by optimization algorithms” describes the identification optimization technique using different methods to simplify the fractional system study and decrease computation complexity. The FOPID controller is present in Section “Fractional PID controller tuning” with the methods used to determine these five parameters. To demonstrate the efficacy of the suggested theoretical notion, two numerical examples of fractional systems will be provided in Section “Results and discussion.” At last, the conclusion is given in Section “Conclusion.”

Fractional order dynamic systems and fractional calculus presentation

Leibniz discussed a potential method for differentiating non-integer orders of m in letters he wrote to J. Wallis and J. Bernoulli in 1695. The definition is as follows:

\frac{d^{m} e^{nt}}{d t^{m}} = n^{m} e^{nt}

(1)

Fundamental operator of non-integer order in fractional calculus is called $t_{0} D^{m} t$ , with bounds $t$ and $t_{0}$ . This is the continuous integro-differential operator, according to Oustaloup^18,19:

{{}_{t_{0}}D}_{t}^{m} = {\begin{matrix} \frac{d^{m}}{dt} R (m) > 0 \\ 1 R (m) = 0 \\ \int_{t_{0}}^{t} {(d τ)}^{- m} R (m) < 0 \end{matrix}

(2)

where the operation’s order, $m$ , usually falls within $R$ . Complex numbers are represented by the ring $τ$ , and their real component is denoted by $R (m)$ .

The fractional integro-differential equation can be expressed in the literature in a variety of ways. Rieman–Liouville approximations are among the most used methods for solving fractional integro-differential equations.

Let $\propto ϵ C$ with $R (\propto) > 0$ , $t_{0} ϵ R$ and let $f$ be a function defined on $[t_{0}, + \infty [$ with a lower constraint $t_{0}$ that is globally integrable. The Riemann–Liouville integral of order $m$ of $f$ is defined by,⁹

{}_{t_{0}}^{RL}{I_{t}^{\propto}} f (t) = \frac{1}{Γ (\propto)} \int_{t_{0}}^{t} {(t - τ)}^{\propto - 1} f (τ) d τ

(3)

where $Γ (\propto$ ) is the Euler gamma function and $t > t_{0}$ .

The derivative of Riemann–Liouville of order $\propto$ of $f$ with lower bound $t_{0}$ is defined as follows:

{}_{t_{0}}^{RL}{D_{t}^{\propto}} f (t) = \frac{1}{Γ (h - \propto)} \frac{d^{h}}{d t^{h}} \int_{t_{0}}^{t} {(t - τ)}^{h - \propto - 1} f (τ) d τ

(4)

where $(h - 1) < \propto < h$ ; $h$ is an integer.

With $r (t)$ serving as the input variable and $y (t)$ providing the output variable, equation (5) is a fractional differential equation that is modeled using the fraction calculus.

D^{\propto} = {{}_{0}D}_{t 0}^{\propto}, D^{β} = {{}_{0}D}_{t 0}^{β}

(5)

$a_{n}, a_{n - 1}, \dots a_{0}$ : constants coefficients of y(t).

$b_{m}, b_{m - 1}, \dots b_{0}$ : constants coefficients of u(t).

$α_{n}, α_{n - 1}, \dots α_{0}$ and $β_{m}, β_{m - 1}, \dots β_{0}$ are commensurate orders ( $α_{n} = aq$ ) are derivative orders that come from integer orders multiple of the order, $q ϵ R^{+}$ .

The order of the procedure can be presumed, without losing generality, to be:

α_{n} > α_{n - 1} > \dots > α_{0}

(6)

and

β_{m} > β_{m - 1} > \dots > β_{0}

(7)

\begin{matrix} a_{n} D^{\propto_{n}} y (t) + a_{n - 1} D^{\propto_{n - 1}} y (t) + \dots + a_{0} D^{\propto_{0}} y (t) \\ = b_{m} D^{β_{m}} y (t) + b_{m - 1} D^{β_{m - 1}} y (t) + \dots + a_{0} D^{β_{0}} r (t) \end{matrix}

(8)

As mentioned in equation (9),²⁰ by utilizing the Laplace transform on equation (8) with zero initial conditions, the fractional order system can be expressed as a transfer function.

\frac{Y (s)}{R (s)} = \frac{b_{m} s^{β_{m}} + b_{m - 1} s^{β_{m - 1}} + \dots + b_{0} s^{β_{0}}}{a_{n} s^{\propto_{n}} + a_{n - 1} s^{α_{n - 1}} + \dots + a_{0} s^{α_{0}}}

(9)

Approximations are important for fractional-order transfer functions. Put another way, when corrector simulations are required, full-order transfer functions are utilized rather than fractional-order transfer functions. Equation (10) shows the Oustaloup^18,19 simple approximation formulation for fractional order differentiator.

s^{α} = K Π_{m = 1}^{M} [\frac{1 + \frac{s}{ω_{z, n}}}{1 + \frac{s}{ω_{p, n}}}], \propto > 0

(10)

where the function’s gain, denoted by $K$ , is maintained constant to yield a unit gain at $(1 rad / \sec)$ . Its poles and zeroes add up to $M$ . At the $nt Ⓟ$ instant, the estimated frequencies of the poles and zeroes are $ω (z, n)$ and $ω (p, n)$ , respectively, and their values correspond to the expected range of $ω l$ (lower frequency) and $w ℏ$ (high frequency) in the system. We use recursive equations to get the approximate frequency range of poles and zeroes. These several parameters are provided by equations (11)–(15).

ω_{z, 1} = ω_{l} \sqrt{η}

(11)

ω_{p, n} = ω_{z, n} γ,

(12)

ω_{z, n + 1} = ω_{p, n}, η

(13)

γ = {(\frac{ω_{h}}{ω_{l}})}^{\frac{\propto}{M}}

(14)

η = {(\frac{ω_{h}}{ω_{l}})}^{\frac{1 - \propto}{M}}

(15)

The approximation method can be used to create fractional order controllers and fractional order systems.

System identification by optimization algorithms

We have used identification from the open-loop response in this paper. This technique attempts to obtain initial estimates of the process by combining data on an approximate process gain, a dominant time constant, and a time delay.

When all other inputs are held constant, the input signal used is a step change of one of the process inputs, the regulated process must be in a steady state before the step changes, and the measured process response needs to be further normalized for unit step changes and initial conditions of zero.²¹ The process is identified by the transfer function as follows:

G (s) = \frac{b_{m} s^{β_{m}} + b_{m - 1} s^{β_{m - 1}} + \dots + a_{0} s^{β_{0}}}{a_{n} s^{\propto_{n}} + a_{n - 1} s^{α_{n - 1}} + \dots + a_{0} s^{α_{0}}}

(16)

Model = \frac{K}{1 + Ts} e^{- Ls} (parameter s number = 3)

(17)

In this approximation method, the model, a first-order system with delay, is obtained by assimilating the step response of these fractional systems to equation (17). However, in order to fine-tune controller parameters, the use of optimization techniques has expanded. The artificial intelligence algorithms²² have exhibited exceptional levels of performance. Researchers generally use empirical data from computer trials when conducting research in the field of metaheuristics. Furthermore, the classification of diverse metaheuristic methods^23–25 is demonstrated in this context by Figure 1.

Figure 1.

Schematic diagram of meta-heuristics algorithms.

We select four algorithms to utilize in our work: The Artificial Bee Colony (ABC),²⁶ The Genetic Algorithm (GA),²⁷ Ant Colony Optimization ACO^{12–21,26–28} and Particle Swarm Optimization (PSO).

The diagram illustrating PSO algorithm is in Figure 2 and the Pseudo code is in Section “Pseudo code of particle swarm optimization (PSO) algorithm.”

Figure 2.

Flowchart of the (PSO) algorithm.

Equations (18) and (19) yield the revised velocity and location, respectively.

v_{ij}^{k + 1} = w . v_{ij}^{k} + c_{1} . ran d_{1} (p_{best} - x_{ij}^{k}) + c_{2} . ran d_{2} (g_{best} - x_{ij}^{k})

(18)

x_{ij}^{k + 1} = x_{ij}^{k} + v_{ij}^{k + 1}

(19)

for which $p_{best}$ is the best location of particle and $g_{best}$ is the best global location in the population. The $C$ 1 and $C 2$ are positive constants. C2 is the representation of the global learning rate, and C1 is the cognitive earning rate.

Pseudo code of particle swarm optimization (PSO) algorithm

Out of the four models for process control that were found using different techniques, The most efficient one is displayed in Figure 3’s block diagram. In this diagram, the excitation input is denoted by r(t), the actual plant, y(t), is represented by a non-integer transfer function, and the simplified model, y*(t), is a higher-order system devoid of fractional components. By utilizing the $ITAE$ between the simplified model $y * (t)$ and the real pressure $y (t)$ , the objective function fobj is determined.

ITAE = \int_{0}^{+ \infty} t | e (t) | dt

(20)

Algorithm: Particle Swarm Optimization (PSO)
Input: – Objective function to optimize, $f_{obj}$
–Swarm size, $M$ & fix dimension,

d

–Search space range,

[minX, maxX]

Output: –Best value found,

x^{*}

Initialization:For each particle

i

in the problem space:
–Initialize position

x_{i} = (x_{i 1}, . . ., x_{id})

–Initialize velocity

vi = (v_{i}, . . ., v_{id})

–Evaluate

f_{obj} (x_{i})

d

variables and set

[pbest]_{i}

–Set

gbest

as the best of

[pbest]_{i}

Repeat: –Compute inertia weight

w

–Apply equation (16) to update

v_{i}

for every particle.
–Apply equation (17) to update

x_{i}

for every particle.
–Evaluate f(xi) in d variables and update

[pbest]_{i}

–If

f_{obj} (xi)

is better than

[pbest]_{i}

, update

[pbest]_{i}

←

x_{i}

– If the best of

[pbest]_{i}

is better than

gbest

,
–update gbest←best of

{[pbest]}_{i}

Until: –Termination criteria (such as maximum iterations or reaching error tolerance) are satisfied:
–Set

[x]^{*}

←

gbest

–Return

x^{*}

End of Algorithm.

Figure 3.

Algorithms-based system identification of process control system.

Four algorithms of optimization will be applied to find the specific model which is a simple first-time system with a delay.

Fractional PID controller tuning

The mathematical formulation for the Fractional Order (FOPID) controller is represented by equation (21), where the values of the derivative and integrator are between 0 and 1. The fractional PID controller’s block diagram, which is organized as follows, is shown in Figure 4:

C_{FOPID} (s) = \frac{U (s)}{E (s)} = K_{p} + \frac{K_{I}}{s^{λ}} + K_{d} s^{μ}

(21)

Figure 4.

Fractional PID controller block diagram.

The FOPID has two parameters more than the classic PID corrector.

As seen in Figure 5, a fractional PID controller turns into a traditional PID controller if $λ = μ = 1$ .

Figure 5.

The plane of the fractional PID controller.

A control loop is depicted in Figure 6’s block diagram. The plants are also present in this loop, and the Fractional Order Proportional Integral Derivative (FOPID) controller models are designated as G(s) and C(s). The objective is to adjust the disturbance signal D(s) about the reference input R(s) while keeping an eye on the output signal Y(s). The corrector receives the error signal E(s) to accomplish this and creates the control signal U(s).

Figure 6.

Closed-loop system with FOPID controller.

Stabilization with Zeigler–Nichoks methods

Using the open loop answer as a starting point for adjustment: The Ziegler–Nichols open loop approach served as the model for this adjustment method, which was introduced by Valerio and Costa.²⁹ The parameters $K_{p}, K_{I}, λ$ , $K_{d}$ , and $μ$ all vary regularly with $L$ and $T$ . It was able to use a least-squares fit to fit a polynomial to the data, allowing for the discovery of (approximate) parameter values by straightforward algebraic calculations. Table 1 has the values.

\begin{matrix} K_{p} = P = - 1.0574 + 24.5420 L + 0.3554 T \\ - 46.7325 L^{2} - 0.0021 T^{2} - 0.3106 TL \end{matrix}

(22)

Table 1.

Parameter’s adjustment of FOPID by the first method based on the open loop response.

	Parameters to use when $0.1 \leq T \leq 5$
Parameter	$P$	$I$	$λ$	$D$	$μ$
$1$	$- 1.0574$	$0.6014$	$1.1857$	$0.8796$	$0.2778$
$L$	$24.5420$	$0.4025$	$- 0.3464$	$- 15.0846$	$- 2.1522$
$T$	$0.3544$	$0.7921$	$- 0.0492$	$- 0.0771$	$0.0675$
$L^{2}$	$- 46.7325$	$- 0.4508$	$1.7377$	$28.0388$	$2.4387$
$T^{2}$	$- 0.0021$	$0.0018$	$0.0006$	$- 0.0000$	$- 0.0013$
$LT$	$- 0.3106$	$- 1.2050$	$0.0380$	$1.6711$	$0.0021$

A secondary technique for adapting based on the closed-loop response: The system is inserted in a closed loop with a particular proportional gain which gives a pumping phenomenon. Valerio and Costa obtained the information needed to determine the parameters $K_{P}$ , $KI$ , $λ$ , $Kd$ and $μ$ which vary regularly with $Kcr$ and $Pcr$ . Table 2 compiles the parameters of the related polynomials.

\begin{matrix} K_{p} = P = 0.4139 + 0.0145 Kcr + 0.1584 Pcr \\ - \frac{0.4384}{Kcr} - \frac{0.0855}{Pcr} \end{matrix}

(23)

Table 2.

The second method’s parameter adjustment for the FOPID, according to the closed-loop response.

Parameter	$P$	$I$	$λ$	$D$	$μ$
$1$	$0.4139$	$0.7067$	$1.3240$	$0.2293$	$0.8804$
$K_{cr}$	$0.0145$	$0.0101$	$- 0.0081$	$0.0153$	$- 0.0048$
$P_{cr}$	$- 0.1584$	$- 0.0049$	$- 0.0163$	$0.0936$	$0.0061$
$1 / K_{cr}$	$- 0.4384$	$- 0.2951$	$0.1393$	$- 0.5293$	$0.0749$
$1 / P_{c r}$	$- 0.0855$	$- 0.1001$	$0.0791$	$- 0.0440$	$0.0810$

Stabilization with optimization algorithms

The identical optimization algorithms (ABC, ACO, GA, and PSO) will be used in the following to ascertain the corrector C(s)’s ideal parameters. Figure 6 shows the closed-loop control system’s design.

MATLAB-SIMULINK is utilized to apply the several techniques that are employed to ascertain the parameters of the corrector FOPID C(s). The structural diagram of the operation is represented in Figure 7.

Figure 7.

Schematic diagram of the process of setting FOPID parameters by optimization algorithms.

A fractional system can be an expression of a variation of process parameters during operation. The most frequent robustness of this variation is on the polynomial coefficients of the transfer function, but it can also be a variation of the values of the powers in a fractional way, for example, s² can change to s^1.8. For this, our study has been devoted to the stability of fractional systems with fractional regulators which are more robust and give more performance.

To create controllers that handle nonlinearities more skillfully, optimization methods can provide improved handling of nonlinearities. The potential of the FOPID corrector to reduce control effort is another crucial factor, particularly if it is tuned by suitable algorithms while maintaining the required performance.

Several performance standards apply to controller design:

Integrated squared error (ISE):

ISE = \int_{0}^{+ \infty} e^{2} (t) dt

(24)

Integral of absolute error (IAE):

IAE = \int_{0}^{+ \infty} | e (t) | dt

(25)

Integral of time-weighted squared error (ITSE):

ITSE = \int_{0}^{+ \infty} t e^{2} (t) dt

(26)

Integral of time-weighted absolute error (ITAE)

ITAE = \int_{0}^{+ \infty} t | e (t) | dt

(27)

Results and discussion

This section’s simulation of two fractional system examples serves as an illustration of the suggested tuning technique.

Example 1:

Equation (28) gives the chosen systems’ transfer function:

G_{1} (s) = \frac{80}{s^{2.8} + 65 s + 100}

(28)

Identification

Figure 8 displays the fractional system $G_{1} (s$ ) step response. The fractional order $α_{n} = 2.8$ given in equation (28) have an influence on the response of the system.

Figure 8.

Step response of G₁(s) in an open loop.

We applied four optimization algorithms (ABC, ACO, GA and PSO) for $G_{1} (s)$ to obtain a Model described by equation (17). The objective function is defined by the integral of the quadratic error ITA. From Table 3, the best curve that presents an almost zero error (0.0045) is the one determined by the PSO algorithm. Also, from Figure 9 it is clear that the transfer function determined with PSO presents the model as close as possible to the real process.

Table 3.

Results for model 1 structure.

Algorithm	K	T	L	Error
ABC	0.8090	0.6782	0.1000	0.0053
ACO	0.7867	0.3551	0.1590	0.0142
GA	0.7922	0.6700	0.1310	0.0116
PSO	0.7985	0.6208	0.1000	0.0045

Figure 9.

Open loop step replies for comparison.

Next, we suggest a straightforward technique to recognize a group of fractional systems possessing a delayed first-order model. We move from a system that presents non-linearities that can be difficult to treat with ordinary stability theorems towards a classical linear system.

The $G_{1} (s)$ process is now equated to a first-order system with delay¹⁵:

G_{1} (s) ≃ \frac{K}{1 + Ts} e^{- Ls} = \frac{0.785}{1 + 0.6208 s} e^{- 0.1 s}

(29)

The stability of fractional systems can now be studied using new theorems.^30,31 Our method is easy to use for stabilizing this particular class of fractional systems. A stability study with a well-designed regulator must follow the identification of the fractional system. We subsequently propose a fractional regulator FOPID. In the same context of fractional calculation, the parameters K, T, and L are useful for calculating the different parameters of the regulator with the extension of the Z–N method.

Stabilization with optimization methods

The optimization of the FOPID parameters with different algorithms (GA, ACO, ANT, or PSO) is proposed with the method illustrated in Figure 10.

Figure 10.

The structure of GA, ANT, ABC, or PSO of the FOPID tuning system.

Six methods of tuning the fractional PID controller have been introduced. The Zeigler–Nickols rules are used to modify the controller’s five parameters, and algorithms are then used to minimize the performance index criterion. The FOPID is obtained using the diffusive representation of the fraction operator, which allows a rational transfer function implemented with Oustaloup’s approximation.

An analysis was conducted to compare the six controllers that were employed to demonstrate the effectiveness of the suggested controllers. The suggested controllers’ FOPID performances (overshoot, rising time, and settling time) were obtained.

It is clear that everyone is stable in this case when we test the system’s stability using all of the intended FOPIDs. However, while comparing the characteristics such as overshoot and settling time, it becomes apparent that these values of overshoot and settling time decrease in additional ways for optimization techniques. We can see from Figure 11 and Table 4 that the settling times were 16.7266 and 17.3147 s, and the overshoot values calculated using the Ziegler–Nichols methods were 35.6483 and 43.6377, respectively.

Figure 11.

Closed loop responses of $G_{1} (s)$ with FOPID: (a) Responses with both ZN methods and (b) Responses with optimization methods.

Table 4.

Controller parameters and performance index of $G_{1} (s)$ .

Method	$K_{P}$	$K_{I}$	$λ$	$K_{d}$	$μ$	$Overshoot (%)$	$Risetime$	Settling time
ZN1	0.3014	0.4464	1.4761	0.1350	0.9803	35.6483	3.4005	16.7266
ZN2	0.1137	0.3673	1.5208	−0.0965	1.0585	43.6377	3.0366	17.3147
ABC	2.1381	2.1382	1.0151	0.7263	1.4618	0.9335	0.0850	1.9269
ACO	1.6805	2.2598	1.0167	0.1261	1.1080	4.3677	0.7825	2.8454
GA	2.9782	3.9978	1.0145	0.3250	1.3936	0.6799	0.9001	1.0516
PSO	2.4951	3.9744	0.9421	0.1000	1.6187	0.0000	0.6380	1.0911

Using optimization algorithms, the overshoot values decrease to 0.9335, 4.3677, 0.6799, and 0.000, the settling time is respectively 1.9269, 2.8454, 1.0516, and 1.0911. So, the fractional system can achieve a steady output level swiftly and without much overshoot. Hence, we can conclude that FOPID controller with optimization algorithms gives stability as well as good step response characteristics, above all with an adequate method.

The adjustment results of the FOPID corrector based on the PSO method show more performance than other algorithms. We can see this by graphic comparison. If we want to specify, we may conclude that the PSO algorithm exhibits a superior control effect because of its quick response, lack of overshoot, low regulation time, and ability to reach the steady-state zone.

Now we need to prove that this PSO optimization method for corrector parameter adjustment also has minimal value control effort and also gives robustness even after model variation.³²

As shown in Table 5 and Figure 12, the control effort required in the two methods of Ziegler–Nichols was 104.12 and 91.23 but with poor performance so it provides little energy to stabilize this split system without any effect on system control. The Y axis is presented in its logarithmic form to simplify reading.

Table 5.

Values of control effort for different control methods.

Corrector	ZN1	ZN2	ABC	ACO	GA	PSO
$‖ u ‖$	91.23	104.12	10⁴	197.15	4 × 10³	3 × 10³

Figure 12.

Control inputs for G₁ (s) with different corrector.

The effort required to have zero overshoot is 3 × 10³ by the PSO method, whereas the ABC algorithm requires an effort equal to 10⁴ for 0.9 overshoot. So, it is much better with the minimum effort we have good performance compared to other optimization methods, and we have a control input value minimized almost by 75%.

Guaranteeing stability, with good performance, while providing as little effort as possible, for split systems is a challenge that we have achieved. Thus, a close examination of the potential implications of our results for future research avenues and real-world applications shows that the well-optimized FOPID corrector is an ideal tool for certain classes of fractional systems. Here are a few actual systems to which we can apply our findings: Viscoelastic materials which are neither strictly elastic nor strictly viscous, and their behavior can be represented using fractional differential equations to represent the response of the material over time. Fractional order models are also utilized in electrochemical systems to explain the dynamics of non-integer order behavior in the system, such as in cycles of battery charge and discharge. Our goal is to broaden the conversation and examine in depth the broader implications of our findings compared to previous research,³³ highlighting their possible influence on the fractional system.

The system’s stability will be lowered by disruptive signals, such as parametric uncertainty and external disruptions. This is the reason that, in the pursuit of robust stability and dependable performance, research is increasingly concentrating on the complete management of the system in the face of external shocks and parametric uncertainties. In a steady state, a constant disturbance injection D(s) is injected, as seen in Figure 6. Figure 13 shows the system response with a FOPID corrector optimized by PSO algorithms with injection disturbance. It confirms that the fractional system $G_{1} (s)$ always remains stable over time which validates the capacity of the regulation system to reject external perturbation. The output signal is disrupted when the coefficients of the denominator of $G_{1} (s)$ are arbitrarily chosen at a specific time. There are two step changes added at t = 2 (s) and t = 4 (s), in that order. The closed-loop responses of the main output are shown in Figure 14. We first start with a variation of ±10% to see the behavior of the system in a closed loop and to know if this guarantees robustness or if it diverges completely. The results are convincing since stability quickly returns. Then the second one tried was for a ±20% variation in parameters, we have a small overshoot but the system always remains in a stable state. It is evident that the tracking control completely satisfies the rapid reaction and oscillation-free overshoot system performance requirements. On the other hand, the primary controller keeps things running smoothly. Figure 14’s control signals demonstrate how well the suggested approach operates displays the systems’ rejection of disturbances. It is demonstrated that when an external disturbance appears, systems managed by a reliable FOPID controller swiftly recover to the desired reference value.

Example 2:

Figure 13.

Robust control of G₁ (s) with disturbance external injection.

Figure 14.

Closed-loop reaction of G₁ (s) to perturbation injecting on its variation parameters and coefficients: (a) Perturbation at 2s and (b) Perturbaton at t=4s.

The transfer function of systems chosen is given by:

G_{2} (s) = \frac{5 s^{1.4} + 22}{s^{3.3} + 11 s^{1.7} + 79 s + 100}

(30)

Identification

The $G_{2} (s)$ system is now equated to a first-order system with delay:

G_{2} (s) ≃ \frac{K}{1 + Ts} e^{- Ls} = \frac{0.8841}{1 + 1.220 s} e^{- 0.1 s}

(31)

Identifying suitable values for the controller is essential for establishing stability in a closed-loop system. However, performance remains significantly subpar, as indicated in Table 6. The challenge arises when seeking a controller, that not only ensures good performance but also minimizes the effort required for improvement. In this analysis, we showcase the PSO method’s superiority over the fractional-order PID. Stabilizing the second system demands an effort on the order of 8.3 × 10³ using the ABC algorithm. In contrast, the PSO algorithm achieves a comparable effort level of 944.8. So, we saved 55% of effort. The graph of the FOPID corrector’s effort utilizing the six previously discussed approaches is displayed in Figure 16 with an exponential distribution representation on the Y-axis display.

Table 6.

Values of control effort for different control methods.

Corrector	ZN1	ZN2	ABC	ACO	GA	PSO
$‖ u ‖$	283.1	251.56	8.3 × 10³	157.77	1.7 × 10³	944.8

Using the four techniques, we can extract the various parameters of the first-order model with delay from Table 7. Figure 15 illustrates this streamlining of the analysis of specific fractional system classes.

Table 7.

Results for model 1 structure.

Algorithm	K	T	L	Error
ABC	0.8028	0.8447	0.3020	0.1303
ACO	0.8570	1.0407	0.1760	0.0757
GA	0.8294	1.2370	0.1160	0.1093
PSO	0.8841	1.2220	0.1000	0.0484

Figure 15.

Open loop response of Model 1.

Following the identification phase, an optimization of the fractional regulator FOPID characterizes the control; Table 8 provides these five parameters. When comparing the performance indices of the PSO algorithm versus the Ziegler–Nichols technique extension, they are exceptionally good. The latter shows a stabilization time of 44 s and a substantial difference of about 35%. The PSO method provides us with zero overshoot and a stabilization time of 2.8 s in response to this. We illustrate the closed-loop system’s behavior and the effectiveness of the newly derived control law in Figure 16.

Table 8.

The effectiveness measure and controller settings of $G_{2} (s)$ .

Method	$K_{P}$	$K_{I}$	$λ$	$K_{d}$	$μ$	$Overshoot (%)$	$Risetime$	Settling time
ZN1	0.5503	0.5806	1.3871	0.2710	1.0484	35.6637	5.8678	44.7774
ZN2	0.5175	0.6418	1.3565	0.3833	0.9724	22.8603	6.5969	23.2380
ABC	1.7702	4.2000	1.0571	1.0460	1.0215	6.2898	2.5572	5.7298
ACO	1.1494	2.4745	1.1384	0.9389	0.7409	14.0990	3.2229	9.2165
GA	3.8674	3.7695	1.0851	1.8664	1.2749	3.2794	2.8862	3.3809
PSO	4.0010	3.9958	1.0802	1.7970	1.0287	3.0919	2.8989	3.4232

Figure 16.

Closed loop responses of G₂ (s) with FOPID: (a) Responses with both ZN methods and (b) Responses with optimization methods.

The non-linearity in the fractional system caused by the non-integer fraction in the model has vanished because of the ranges on the derivative and integral of the fractional order of the suggested FOPID controller. The FOPID controller optimized by a suitable algorithm is therefore very effective, especially for systems identified by fractional models, as an implication of these results. This means that even if we increase the range of fractions in the system model (s^3.3 and s^1.7) compared to Kothari et al.,³⁴ Nassef and Rezk,³⁵ and Benftima et al.³⁶ the regulation by the proposed controller remains a better solution due to its simplicity and efficiency (Figure 17).

Figure 17.

Control inputs for G₂ (s) with different corrector.

A simulation of a closed-loop system with an external disturbance injected at t = 8(s) and a regulator chosen using the PSO algorithm is shown in Figure 18. In spite of outside disruptions, the system remains stable. Thus, our techniques demonstrate the usefulness of using a fractional corrector to stabilize a fractional system. The purpose of this paper is to show that it is possible to examine the effects of rejecting disturbances while the servo system is operating.

Figure 18.

Robust control of $G_{2} (s)$ with disturbance external injection.

The denominator’s initial condition is set to $[a_{3} a_{2} a_{1} a_{0}] = [1 1 1 7 9 1 0 0]$ , the system is simulated over 6 s. A disturbance of ±10% (of the denominator value) is applied at t = 2 s and another disturbance of ±20% at 4 s with PSO to analyze disturbance rejection and restore performance. The closed-loop model’s regulatory response is displayed in Figure 19. The FOPID created using the suggested method in this publication matched well against disturbances. It is clear that the system is kept stable by the control law, it quickly returns to its stable position with minimum energy dissipation with disturbance of ±10%. In the case of ±20% disturbance, the system returns to its stable position after a peak and then a small overshoot, which implies that even if we have reached ±20% disturbance, the system remains stable and keeps this performance well. Interestingly, the variation analysis also improves the FOPID robustness study for fractional systems. One of the main outcomes of the proposed FOPID is its resilience under various settings. That is to say, the system maintains stability for a ±10% perturbation injection or even a higher ±20% perturbation injection.

Figure 19.

Robust control of $G_{2} (s)$ with disturbance-induced coefficient variations: (a) Perturbation at 2s and (b) Perturbaton at t = 4s.

It is imperative to underscore that our research on fractional systems has introduced a novel FOPID topology that exhibits greater resilience than earlier studies documented in the literature. Our search on the Engine Control System can be used to improve fuel efficiency and performance. In order to capture the non-linear and memory effects inherent in the system, fractional-order systems can be used to represent the hydraulic and mechanical components of the brake system, giving our controllers greater tuning flexibility and durability. Previous works’ main drawbacks are that they only used one optimization approach, didn’t do a robustness analysis or find optimal solutions, overshot, or had slow stabilization durations.

Conclusion

We present a novel method for handling a certain class of fractional system. The identification consists of assimilating the fractional system to a simplified first-order model with delay. We have developed optimization algorithms to identify this model. This model allows us to stabilize the real fractional process with the FOPID tuning methods calculated by Ziegler–Nichols. To improve performance, particularly from the point of view of overshoot and stabilization time, we have also designed a control law using these algorithms, which gives us good results in PSO optimization compared to other methods. The system controller performs effectively while minimizing the energy required for control efforts. Quantitatively, the savings in effort depend on the specific system being addressed; in our cases, it ranges from approximately 75% to 55%. The simulation was applied to two fractional systems with 10% parameter variation to show the robustness of the developed controllers.

Footnotes

Acknowledgements

We would like to express our sincere gratitude to Dr Achraf Jabeur Telmoudi my Supervisor for their invaluable guidance, support, and expertise throughout the course of this research project. Additionally, we extend our appreciation to the entire laboratory team for their collaboration, dedication, and contributions, which were instrumental in the successful completion of this study: Dr Abdelkader Chaari and Dr Adel Taieb.

Author contributions

All authors contributed to the study conception and design, data collection and analysis were performed by Bilel Kanzari, Adel Taieb, Abdelkader Chaari and Achraf Jabeur Telmoudi. The first draft of the manuscript was written by Bilel Kanzari and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Bilel Kanzari

Data availability statement

Within the scope of this study, all utilized data and relevant materials are available upon request from the authors.

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Intelligent-based approach for parameters identification and control of fractional order systems

Abstract

Keywords

Introduction

Fractional order dynamic systems and fractional calculus presentation

System identification by optimization algorithms

Pseudo code of particle swarm optimization (PSO) algorithm

Fractional PID controller tuning

Stabilization with Zeigler–Nichoks methods

Stabilization with optimization algorithms

Results and discussion

Identification

Stabilization with optimization methods

Identification

Conclusion

Footnotes

Acknowledgements

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

Data availability statement

References