Ingeniería e Investigación
0120-5609
Facultad de Ingeniería, Universidad Nacional de Colombia.
https://doi.org/10.15446/ing.investig.106657

Recibido: 6 de enero de 2023; Aceptado: 24 de julio de 2023

Reduction of Large Scale Linear Dynamic MIMO Systems Using ACO-PID Controller

Reducción de sistemas MIMO dinámicos lineales a gran escala mediante el controlador ACO-PID

J. Daif-Alkhasraji, 1 S. Shneen, 2 M. Sulttan, 3

1 Electrical engineer, University of Technology (UOT), Iraq. PhD in Optical Wireless Communication Engineering, Newcastle University, United Kingdom. Affiliation: Lecturer, University of Technology, Electromechanical Engineering Department, Baghdad, Iraq. E-mail: jaafar.m.dhaif@uotechnology.edu.iq Shahrood University of Technology University of Technology Electromechanical Engineering Department Baghdad Iran
2 Electrical engineer, University of Technology (UOT), Iraq. PhD in Power Electronics, Huazhong University of Science and Technology (HUST), China. Affiliation: Energy and Renewable Energies Technology Center, University of Technology, Baghdad, Iraq. E-mail: salam.w.shneen@uotechnology.edu.iq Shahrood University of Technology University of Technology Baghdad Iran
3 Electrical engineer, University of Technology (UOT), Iraq. PhD in Information and Communications Engineering, Huazhong University of Science and Technology (HUST), China. Affiliation: Assistant professor, University of Technology, Electromechanical Engineering Department, Baghdad, Iraq. E-mail: mohammed.q.mohammed@uotechnoloy.edu.iq Shahrood University of Technology University of Technology Electromechanical Engineering Department Baghdad Iran

ABSTRACT

The MIMO technique is an essential element in the standards of communication systems (IEEE 802.11n, IEEE 802.11ac, WiMAX, and LTE) because it helps to increase their capacity. This paper employs a model order reduction (MOR) technique with a PID controller, an ACO algorithm, and the ITAE fitness function to reduce the large-scale linearity of the MIMO technique. The numerator and denominator parameters are set by minimizing the ITAE fitness function between the transient responses of the original and the reduced model. The objectives are achieved with the PID controller and the ACO algorithm for the unit step input. The simulation results show a good system performance. The controller performance is presented with regard to the dynamic response in terms of rising time, settling time, and overshoot/undershoot. Moreover, the results of the proposed method are compared with four literature reports for validation purposes. Evaluating the parameters within the time frame and the error values with and without the PID controller and ACO algorithm allowed validating the functioning of the proposed method. Furthermore, the simulation results revealed that the proposed scheme exhibited sufficient robustness and demonstrated a reduction in the time-domain response and error values.

Keywords:

LS-MIMO, model order reduction (MOR), PID controller, ant colony optimization.

RESUMEN

La técnica MIMO es un elemento esencial en los estándares de sistemas de comunicación (IEEE 802.11n, IEEE 802.11ac, WiMAX y LTE) porque ayuda a aumentar la capacidad del sistema. En este trabajo se utiliza una técnica de reducción del orden del modelo (MOR) con un controlador PID, un algoritmo ACO y la función fitness de ITAE para reducir la linealidad de gran escala de la técnica MIMO. Los parámetros del numerador y denominador se establecen minimizando la función de aptitud de ITAE entre las respuestas transitorias del modelo original y el modelo reducido. Se alcanzan los objetivos con el controlador PID y el algoritmo ACO para la entrada de paso unitario. Los resultados de la simulación muestran un buen rendimiento del sistema. El rendimiento del controlador se presenta con base en la respuesta dinámica en términos de tiempo de subida, tiempo de asentamiento y sobreimpulso y subimpulso. Además, los resultados del método propuesto se comparan con cuatro reportes de la literatura para su validación. La evaluación de los parámetros en el marco temporal y de los valores de error con y sin el controlador PID y el algoritmo ACO permitió comprobar el funcionamiento del método propuesto. Asimismo, los resultados de la simulación revelaron que el esquema propuesto presentaba suficiente robustez y demostraba una reducción de la respuesta en el dominio temporal y de los valores de error.

Palabras clave:

LS-MIMO, reducción del orden del modelo (MOR), controlador PID, optimización por colonias de hormigas.

Introduction

Enhancing spectral efficiency is essential, especially in 5G wireless communication systems. In this regard, large-scale systems with multiple inputs and multiple outputs (LS-MIMO) are a promising technology that supports a broad notch of the data for multiple concurrent users, along with the ability to increase spectral efficiency. Furthermore, high-order MIMO can considerably improve the dependability and security of communication systems, obtaining extraordinary locative multiplexing gains, unlike restricted range systems (Ouyang & yang, 2018; Sulttan, 2019; Han et al., 2020; X. Yang et al., 2020; Dang et al., 2021). Moreover, by increasing the physical antenna size, many intrinsic advantages can be exploited, e.g., channel hardening and increased cell coverage, area, and throughput. This is allowed by high-order MIMO in integration with massive infrastructures such as airport roofs, the walls of theaters and stadiums, and large indoor shopping centers, also providing an extremely reliable technology for 6G systems (Khudhair et al., 2016; Sulttan, 2016). Despite the benefits of LS-MIMO, these advantages come at the price of increasing the complexity of the hardware, which is most notable in signal processing operation for transceivers. Therefore, there is still a need to seek an additional pragmatic, low-complexity reducing algorithm for large-scale systems (Rusek et al., 2012). It is noteworthy that, for point-to-point configurations, the receiver aspect has additional complexity when compared to the transmitter side. For example, the more antennas are employed for the transmitter aspect, the higher the system complexity, which rises exponentially with a huge number of antennas.

On the other hand, extra attention is paid to the receiver aspect of the MIMO model, as the transmitted data must be provided to several users (Jaldén & Ottersten, 2005; Larsson, 2009). In terms of applications, MIMO has been considered to be one of the crucial technologies in third, fourth, and fifth-generation (3G, 4G, and 5G) wireless communication systems since it offers considerable increases in link range and data rate without additional bandwidth or transmit power increases. Such technologies play an essential role in many applications in communication systems, such as GSM, Bluetooth, WiMAX, WLAN, wireless sensors, portable devices, handheld gadgets, automobiles, smartphones, and many more (Sharma et al., 2022; Bizaki, 2011). Furthermore, in order to provide an acceptable service, the number of antennae must be increased for multi-user scenarios. In this vein, large arrays come with a crucial challenge regarding reliability and computationally effective detection (Lyu et al., 2019). As for analysis or controller design, complicated expressions to simulate physical techniques, which imply complex handling of high-order differential formulas, have inevitably led to low-order designs. Such mathematical formulas could be adopted to provide a sufficient performance. Hence, comprehensive knowledge is required, along with decreasing complexity with simple system designs (Vasu et al., 2012). A considerable amount of literature on designing PID controllers for SISO and MIMO prototypes has been published (Lengare et al., 2012; Labibi et al., 2009; Rajinikanth & Latha, 2012; Hu et al., 2010). One of the critical missions of LS-MIMO technology is to reduce the complexity of dynamic designs. Given that dynamic systems contain many variables, notably many mathematical models, it is hard to adopt a complex mathematical form, since the system works in real time. Additionally, memory capacity requirements must be fulfilled, and quick and reliable results must be obtained based on system design (Juneja & Nagar, 2015; Al-doraiee & Al-Qaraawi, 2013; Muhsen & Raafat, 2021). Therefore, over the past few years, heuristic techniques have been widely employed in the literature to achieve optimal performance regarding the obstacles posed by MIMO controller configurations (Hassanzadeh & Mobayen, 2011; Lones, 2014). The difficulty of controlling the MIMO scheme arises from the disruption occurring in one branch, which entails changes in the other branches. Compared to the traditional methods, such as the Zeigler and Nichols and Cohen and Coon tuning techniques, the heuristic process is free of any mathematical calculations. Much of the current literature on MIMO proposes several heuristic approaches to obtain an optimum performance via tuning approaches, e.g. , Genetic Algorithms (GA), Particle Swarm Optimization (PSO), Bacterial Foraging Optimization (BFO), the Firefly Algorithm (FA), and Ant-Colony Optimization (ACO) (Saroja et al., 2017; Sivakumar et al., 2015; X.-S. Yang, 2009; Daraji & Hale, 2014; AL-Suhail & Miry, 2015; Aziz et al., 2022). It is common knowledge that organizing and executing a controller-based centralized method is a complicated issue when compared to the decentralized method. It is worth adding that, according to the criteria for designing MIMO systems, several methods must be adopted, i.e., the centralized method (Vu et al., 2007), the decentralized method (Lengare et al., 2012; Lones, 2014; Labibi et al., 2009), and optimal controllers with decouplers (Dhanraj & Nanjundappan, 2014; X.-S. Yang, 2009; Liu et al., 2003; Sivakumar et al., 2015).

Suman and Kumar (2020) suggested an approach to reducing the order of large-scale linear dynamical systems based on the Balanced Singular Perturbation Approximation method. This method relies on retaining the dominant system modes and comparatively eliminating the less significant eigenvalues. Furthermore, through the proposed approach, the time and frequency response of the system has shown an excellent match when compared to that of other approaches in the literature. Kumari and Vishwakarma (2019) proposed an algorithm that employs the renovated pole clustering technique and numerator coefficients computed by factor division in order to reduce the order of large-scale linear dynamic MIMO systems. This method is also applicable to linear MIMO systems. Abdullah (2016) proposed a new modified PSO technique to reduce the order of large-scale linear SISO and MIMO systems based on the linear descending inertia weight strategy. The suggested modification of the PSO algorithm provided better characteristics in terms of global search capabilities and fast convergence speed, thus making it effective and efficient. Abu-Al-Nadi et al. (2013) proposed a model order reduction using an invasive weed optimization technique in a linear multivariable system. This technique was applied with the combined advantages of retaining the dominant poles and error minimization. The proposed algorithm showed good results, reducing the 10th order of the MIMO linear model in a practical power system to the 3rd order. vishwakarma and Prasad (2009) suggested a new mixed method to reduce the order of large-scale linear dynamic MIMO systems. This method uses modified pole clustering to synthesize the common denominator polynomial of the reduced-order transfer function matrix while a GA computes the coefficients of the numerator elements by reducing the integral square error between the time responses and the system elements. The proposed algorithm was illustrated and compared against other reduction techniques, showing good stability when the original high-order system is stable.

In this manuscript, the MIMO PID control is optimized while adopting the ACO algorithm, and it is integrated with ITAE as a fitness function. Two scenarios are analyzed, i.e., open and closed loops with and without the PID controller, aiming to achieve system robustness and reduce the order of the large-scale MIMO system.

PID controller

The most straightforward and attractive solution in industrial control design implementation corresponds to Proportional-Integral-Derivative (PID) controllers, which do not have a complicated configuration and are very efficient. This, in addition to their competitive pricing and lowmaintenance expenses. Furthermore, through adequate parameter tuning, the dynamic response of a system can be improved by reducing the overshoot while reducing and/or eliminating the steady-state error, consequently enhancing the system's stability (Hanifah et al., 2013; Nagaraj & Murugananth, 2010; Xu,2010).

The purpose of tuning a PID controller is to ensure that its proportional, integral, and derivative factors are appropriate to the design, aiming to guarantee a closed-loop scheme implementation and a strong arrangement of the control loop over a broad spectrum of functions (Xu, 2010). It is worth mentioning that it is hard to adjust the PID controller without expertise. A manual optimum adjustment could be easy for experts because of the long times needed to obtain ideal operation results. In this regard, every factor in the PID controller could yield different outcomes. For instance, when adjusting to obtain a more satisfactory transient design response, the model response might be delayed under disruption circumstances (Hanifah et al., 2013; Nagaraj & Murugananth, 2010). Such a controller for SISO (Figure 1a) can be expressed mathematically in the time domain as in Prainetr et al. (2019); Oudah et al. (2021).

where k p , k ¡ , and k d represent the proportion gain factor, the integral gain factor, and the derivative gain, respectively, while e stands for the error and t denotes simultaneous time.

Block diagram of SISO and MIMO controllers

Figure 1: Block diagram of SISO and MIMO controllers

Source: (Prainetr et al., 2019; Oudah et al., 2021)

In the s-domain, the transfer function of Equation (1) takes the form of Equation (2) (Saad et al., 2012).

where E (s), U (s) represent the error and control signals applied to the plant model, respectively. For convenience, it is said that using two SISO schemes to construct a MIMO system is more applicable. Hence, this research adopts a heuristic strategy integrated with PID controller design and the ACO method for considering MIMO approaches. Figure 1b shows a schematic diagram configuration that includes the pair of controllers PID1 and PID2. Such a scheme yields the following expressions (Suresh et al., 2017):

Ant Colony Optimization (ACO)

Scientists have studied the social behavior of insects and animals and simulated them using algorithms. One such technique in the swarm intelligence family is Ant Colony Optimization (ACO) (Hanifah et al., 2013; Nagaraj & Murugananth, 2010; Xu, 2010). This approach has different applications, notably in industrial systems. It was developed by (1991, 1996; ?) and inspired by an empirical implementation conducted by Goss et al. (1989) to characterize ant behavior when it comes to using the shortest trajectories to travel over a binary bridge, which was designed so that one path was longer than the other. The insects were allowed to randomly select the path, with identical ant numbers in each branch. In his experiment, as time went by, it was found that the number of ants using the shortest trajectory was greater than those using the long lane. The reason for this is that, in order to return to the nest earlier, the ants must follow the short route. This, in turn, leads to an extra focus of the pheromone on the fastest time track (Goss et al., 1989).

Figure 2 (Hanifah et al., 2013) depicts the proposed PID controller with ACO. It is noteworthy that all the quantities of each factor were allocated to three different vectors. Moreover, it was assumed that the ants used all vectors as trajectories between the nests by selecting a lane between the terminations (Start-End).

Pictorial description of PID tuning employingACO for a three-parameter controller

Figure 2: Pictorial description of PID tuning employingACO for a three-parameter controller

Source: (Hanifah et al., 2013)

When using the ant algorithm, the pheromone is regularly updated at the start of each trajectory between the three nests, aiming to achieve the fastest round with the lowest expense. This work adopted ACO as the tuning approach for the PID controller. In this context, an arbitrary initial value is set as a pheromone for each junction point. The ants then arbitrarily decide the route they want to follow. Additionally, a number of ants are positioned at the start node and at every single iteration in order to determine their solution to the problem, which is known as a trip. Using pseudorandom proportional action choice rules, the probability P ij in Equation (4) is computed as the estimated value between two nodes i, j according to two parameters: heuristic and metaheuristic (Prainetr et al., 2019).

where T ij stands for the pheromone initial value and is updated regularly according to two parameters: α and ß; and η represents the reciprocal of the span between the two nodes. The value (η ij = 1/d ij ) directs the allocation of ants to the nodes, preventing them from being excessively far away. The impact of pheromone density on a certain route refers to the direction that the ant adopts, denoted by a, which is a positive value in Equation (4). It is noteworthy that, if α = 0, the closest node will be selected every time, whereas, if β = 0, only the chemical factor (pheromone) tracks recreate on the selection (Prainetr et al., 2019; Chebli et al., 2018).

ITAE

iTAE stands for integrating the multiplication of time by an absolute error. The main feature of this approach is that it has acceptable damped oscillationwitha fairlylowovershoot value. The expression (5) defines such a method, while Figure 3 exhibits its block diagram.

Simulation model showing ITAE with a PID controller

Figure 3: Simulation model showing ITAE with a PID controller

Source: (Prainetr et al., 2019; Chebli et al., 2018)

Methodology

In this section, the procedures of the ACO algorithm are presented in some detail with the aid of the flowchart shown in Figure 4 (Prainetr et al., 2019; Chebli et al., 2018).

Flowchart for tuning the PID algorithm via ACO for SISO as a single branch of a MIMO scheme

Figure 4: Flowchart for tuning the PID algorithm via ACO for SISO as a single branch of a MIMO scheme

Source: Authors

According to this Figure, the first step is to assign the initial values of the number of ants, chemical factors, and control parameter constants, suchas K p , K i , and K d , using a random variable uniformly distributed over (0, 1). Then, several ants are positioned at the node, and the values related to the multi-goals is calculated. Afterwards, the consecutive node is selected, and its probability is estimated via Equation (6).

where j = (p, i, d) is investigatory, and denotes path impacts of ant q within a specific interval. In the next step, the value of chemical factor Δтq ij (t) for any iteration t on each route is determined via Equation (7).

where L q (t) signifies the total length traveled by the ant q, and L mln represents the optimization obtained by a group of ants up to the current iteration.

Note that a high value of a might lead the formula to quickly converge to the sub-optimal route by increasing the chemical factor, particularly regarding the initial arbitrary value. For the sake of convenience, the chemical factor concentration was allowed to evaporate in order to prevent early convergence. This, in order to avoid an unlimited increment in chemical factor trails. In other words, a reasonable trade-off between these two factors is required. Thus, the chemical factor is given by Equation (8).

where m denotes the amount of ants, t denotes the iteration, and p indicates the rate of chemical factor vaporization in the range of 0 < p ≤ 1. Afterwards, as soon as the steady-state zone of the PID controller is solved, the optimal quantities must be explored over a limited zone according to the multiparameters.

It is well known that closed-loop schemes are commonly evaluated in terms of resilience, preciseness, reliability, and steady-state transition response, to name a few. In this vein, modern technologies demand additional advanced performance criteria, so the time at which errors occur must be considered. This issue has been widely explained in the literature.

Concerning the system's dynamic, this work considers a remarkable performance criterion known as the Integrated Time Absolute Error standard (ITAE). This index is widely used since it has higher selectivity when compared to other methods. It is less concentrated on the initial error and prioritizes the system error.

It is worth mentioning that this work was conducted using a Simulink model (MATLAB 2018b). This model entrusted with the the transfer function (TF), the ACO algorithm, and the remarkable performance criterion (ITAE) using codes of the M-file type. The PID controller was also implemented using the Simulink model.

Simulation results and discussions

This section presents and discusses the simulation results. This is done in two directions. The first one is to offer and discuss the results of the open-loop system, comparing them against to those of a closed-loop system with and without control (PID controller and ACO-PID controller). Figure 5 shows the step response of the system in an open loop and in a closed loop.

Step response of the system in open and closed loops (with and without control)

Figure 5: Step response of the system in open and closed loops (with and without control)

Source: Authors

The second direction concerns the results obtained for the system in a closed loop without control and in a closed loop with control (PID controller and ACO-PID controller) Figure 6 shows the step response of the system in a closed loop (with and without control). Of course, all types of the system referred to earlier include the cost function ITAE actions which help to reduce the large scale of a MIMO system.

Step response of the closed-loop system with and without control (PID controller and ACO-PID controller)

Figure 6: Step response of the closed-loop system with and without control (PID controller and ACO-PID controller)

Source: Authors

According to the results shown in Figure 5, there is a large performance gap between the open- and closed-loop systems, as there is no optimization in the former, which is also less reliable. On the contrary, the closed-loop system is optimized and more reliable.

The results of the system without control (Figure 6) show the following: rising time = 0,147 s, overshoot = 0,446%, and undershoot = 1,98%. The parameters of the system with a closed-loop PID controller have K p = 2, K = 12, and K d =0,001. Thus, the results are as follows: rising time = 0,0572 s, settling time = 0,418 s, overshoot = 7,27%, and undershoot = -0,003%. Furthermore, the results obtained for the closed-loop system with the ACO-PID controller, which works with K p = 4,8, K i = 9,5, and K d = 0,11, are as follows: rising time = 0,0441 s, settling time = 0,115 s, and overshoot = 0,132%.

In light of the above, the closed-loop system with the ACO-PID controller responds quickly and is relatively stable (i.e., there is a reduction in the large scale of the MIMO system) in comparison with the system with the PID controller and that without control.

Table 1 compares the response parameters of the proposed method based on step input regarding rising time (s), settling time (s), and overshoot (%) in four ways that have already been described in the literature (Goyal et al., 2019; Saraswat et al., 2015; Parmar et al., 2007; Prasad & Pal, 1991). It can be seen that the combination of ITAE and the ACO-PID controller reduces the response time, effectively reducing the large scale of MIMO techniques.

Table 1: Comparison between the proposed method and four techniques introduced in the literature based on step input response

Source: Authors

Conclusion

This work proposes a control method based on the combination of ITAE as a fitness function and ACO as an optimization algorithm with a PID controller. Using MATLAB 2018b, the m.file was dedicated to the transfer function (TF), the ACO algorithm, and the ITAE. Simulink was used to concstruct the PID controller. The desired response model was obtained by changing the three PID parameters (K p , K i , and K d values). The method was validated by measuring time domain parameters such as rising time, settling time, and overshoot/undershoot. According to the simulation results for two scenarios (with and without control), it was concluded that improvements of about 30 and 29,5% can be achieved for response speed and relative stability, respectively. By comparing the results with those in the literature, it can be observed that our proposed approach can provide a remarkable and superior dynamic response. Therefore, it is inferred that this enhanced ratio can reduce the time response of large-scale MIMO systems.

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CRediT author statement

All of the authors have significantly c ontributed t o the research. Jafaar M. Daif-Alkhasraji was involved in the background research, method development, simulation results, drafting of the paper, and results discussion. Salam W. Shneen contributed to the method proposal, the simulation results, and the data collection and tabulation. Mohammed Q. Sulttan provided guidance during the entire research and the writing process. He supervised the research and provided critical feedback.