Optimal and Robust FOPID Controller for Overhead Cranes Systems Based PSO

Overhead cranes serve as vital infrastructure in today's industrial settings, allowing for the movement of very large loads in all three spatial dimensions while also increasing safety and productivity within operations [1, 2]. Overhead cranes, considered mechanical systems, are largely used in production facilities, distribution centers, shipyards, and warehouses—a common trait across these settings is the way that these systems can move loads as controlled three-dimensional motion. The ease or ability to control three-dimensional motion of loads can significantly affect overall productivity and worker safety. At the same time, the operation and control of overhead crane systems represent a significant technical challenge because of its structure. Most phenomena of motion of an overhead crane constitute underactuated mechanical systems—in which the number of control inputs is less than the total number of degrees of freedom for the system [1, 3]. The primary control objectives will remain the same in industrial settings or applications: the achievement of precise positioning, the inhibition of payload sway during motion, and the rejection of disturbances during operation. Historically, crane systems have been controlled using techniques founded on the proportional-integral-derivative (PID) control architecture. Because they are simple to implement, relatively low-computational, and have established tuning rules, traditional PID controllers are still largely used in industrial practice [4, 5]. However, the fixed gain properties of conventional PID controllers limit their capabilities to handle the sometimes-extreme nonlinear dynamics often inherent in underactuated tap systems. Recent work has shown that using meta-heuristic optimization methods, and in particular particle swarm optimization (PSO) algorithms, can mitigate oscillations through explicit tuning of the PID control gain to stabilize payload motion [4]. Model predictive control is a significant advancement in overhead crane control theory, and represents significant theoretical and practical advantages to enable the simultaneous objectives of load stabilization and damp vibrations [1]. Specifically, the MPC framework gives users the ability to move the ball overhead crane system multiple axes in a coordinated motion while minimizing payload oscillations by physically constraining the controller formulation, optimizing over a limited prediction horizon, and systematically predicting future system dynamics for optimal control inputs. Fuzzy Logic Control has arisen as a potent approach for addressing uncertainty and nonlinearities that characterize crane systems that operate under variable load and varying conditions [6]. It has been shown through both simulation and experimental work that hybrid-type control structures that use fuzzy logic with known proportional-integral-derivative control (i.e., fuzzy-PID control) have significantly improved swing suppression capabilities better than traditional PID or pure fuzzy control strategies [7]. Sliding Mode Control techniques have been shown as a robust control methodology for many complex system systems like robotic and crane system with significant external disturbance and model uncertainty [8, 9]. The theoretical merits of SMC, including guaranteed finite-time convergence, less sensitivity to parameter change, and insensitivity to matched external disturbances, have led to authors turning their work toward creating advanced forms that specifically match the conditions of crane systems [10, 11]. Nonlinear, adaptive and fractional-order sliding mode controllers have demonstrated to be very effective managing dynamic and multi-degree-of-freedom crane system models [12, 13]. Recently, there has been extensive research interest in hybrid control architectures, which automatically combine multiple control methodologies. Hybrid control architectures utilizing sliding mode controller mechanisms with PID logic take advantage of the disturbance rejection properties of SMC while preserving the simplicity of implementation [14]. Control systems that incorporate a neural network have proven to outperform PID methods in simultaneous system stability and robustness, as well as oscillation suppression, during dynamic crane motion [15]. In the study [16], a state space fuzzy controller used to represent the nonlinear dynamic of the crane system with the linear-matrix-inequality A Takagi-Sugeno fuzzy control with Input Shaping method are combined to reduce the swing by representing the nonlinear system as a sub system for different operating points [17]. Although, most these presented achieved good performance but in practical, there are many difficulties in implementing these controller. In this paper, FOPID controller with multi object function based PSO algorithm has been suggested. The main contribution of his paper is; use FOPID controller for the crane system, which provide more flexibility and high robustness with respect to PID controller Multi objective optimization technique based PSO has been used to tune the parameter s of the FOPID ensuring good tracking performance with high robust against external disturbance and system uncertainty by selecting suitable weighting function for each term used in the cost function. A novel multi-objective function is designed that taking into account the accuracy, sway angle elimination, transient specifications, and robustness. This proposed performance index is expected to give superior result comparing with the standard performance index like IAE, ITAE, or ISE. Examine the performance of the proposed controller in several cases to check its performance and robustness and compare its results with other standard controllers.

This section investigates the two objective that represents the core of this research. Design a FOPID controller for the Overhead cranes systems, then improving the performance of FOPID controller by using optimization algorithm to tune their gain values.

3.1 FOPID controller design

This section discusses the FOPID, which is represents, an improved version of the standard PID controller. In addition to the three standard parameters of the PID (K_p, K_i, and K_d), two parameters had been added (λ, μ). The control law of FOPID controller can be expressed as follows:

$u(t)=K_p e(t)+K_i \frac{d^{-\lambda}}{d t^{-\lambda}} e(t)+K_d \frac{d^\mu}{d t^\mu} e(t)$    (11)

where, e(t) refers to the difference between the desired response and the measured response. Figures 1 and 2 shows implementation of the proposed control law. The important challenge in this design is how select the gain values of the FOPID to achieve optimal performance. Thus, PSO optimization algorithm had been used to tune these parameters.

Figure 1. Step response for cart position

Figure 2. Step response for payload angle (θ)

3.2 Multi-objective optimization problem

To adjust the gains of the Fractional-Order PID (FOID) controller based on the developed dynamic model of the overhead crane system, PSO was used. In this optimization procedure, PSO will augment an initial swarm of particles in the search space, where particles denote feasible solutions. The position and velocity of particles as matrices are defined by the dimension of the swarm size, or total number of particles used in the search. The position and velocity of each particle are updated according to the following formulas:

$\begin{array}{r}v_i(t+1)=w v_i(t)+c_1 r_1\left(p_i^{\text {best }}-x_i(t)\right) +c_2 r_2\left(g^{\text {best }}-x_i(t)\right)\end{array}$    (12)

$x_i(t+1)=x_i(t)+v_i(t+1)$    (13)

where, $v_i$ and $x_i$ represent the velocity and position of particle at iteration $i . c_1$ and $c_2$ denote Cognitive and Social coefficients respectively. $w$ is Inertia weight. $r_1$ and $r_2$ are random numbers. $p_i{ }^{\text {best}}$ represents the best position of particle i. $g^{\text {best}}$ denotes the global best position. As the algorithm iteratively updates the position and velocity of particles, encouraging an optimal set of controller parameters that generate improvements to the system's performance, the goal becomes to minimize a performance index in the time domain, leading to a faster and more stable system response behavior. The FPID feedback controller developed is designed to ensure that the cart reaches its required final position while minimizing the swing angle of the payload, thereby improving the stability and responsiveness of the overhead crane system, the following optative function used for tuning the gain parameters of the FOPID:

$j=\sum_{i=1}^5 w_i f_i$    (14)

where, $w_i$ denotes the $i$-th weighting function and $f_i$ represent the i -th objective function where $\mathrm{i}=1,2,3,4,5$.

$f_1=\int_0^{t f} t e(t) d t$    (15)

This objective function denotes the integral time square error aims to minimize the difference between desired reference and the measured output.

$f_2=S S$    (16)

This objective function tries to minimize Steady-State used to penalize deviations from the final desired values.

$f_3=M_p$    (17)

This objective function tries to minimize the overshot to ensure minimum deviation from the desired reference.

$f_4=O S C$    (18)

This function minimizes the oscillation magnitude system to reduce undesirable vibrations in the system.

$f_5=\int\left\|T_{z w}(s)\right\|_{\infty} d w$    (19)

This function is $H_{\infty}$ (H-infinity) norm. $\left\|T_{z w}(s)\right\|$ represents the transfer function from the disturbance to the output.

$\left\|T_{z w}(s)\right\|_{\infty} \stackrel{\text { def }}{=} \sup \sigma\left(T_{z w}(j w)\right)$    (20)

The $H_{\infty}$ refers to the maximum possible gain of the controlled system from the disturbance signal to the performance output over all frequencies.

The research presented in this paper is a comprehensive comparative study of linear control strategies for anti-swing control in underactuated overhead crane systems, including a FOPID controller, a standard PID, an analog PD, and a LQR controller proposed in this work and optimized using PSO. The testing shows that the FOPID controller performed best in all our metrics, including reaching the least Integral Absolute Error (IAE = 0.55), no overshoot in nominal conditions, and showing the highest robustness to external disturbances and parameter uncertainty within the parameters prescribed in this paper. The FOPID controller's framework in fractional-order calculus allows an enhanced tuning capability with two additional tuning parameters (λ and μ), and can be tuned using nonlinear crane dynamics much more easily than the conventional controllers that are static in the left-hand plane. In particular, within the disturbance rejection and system uncertainty conditions, the FOPID showed stable performance in nominal conditions with only a slight degradation in performance, while the other controllers showed varying degrees of performance degradation. In conclusion, the FOPID controller is the right candidate for real-world applications of overhead cranes operating under precision positioning and swing take down of the payload, and importantly makes an important contribution towards the controlling of underactuated mechanical systems and also lays a fundamental framework for future exploration of more advanced crane control and uses in automation systems in industry.