Recently, significant works have been presented to design an automated control system of the drug infusion system to handle challenges in drug infusion systems, such as time-varying sensitivity, complexity, and the existence of measurement noise. A classical proportional-integral-derivative PID controller has been applied in [7], where nonlinear PID digital control is presented in [8]. Also, traditional rule-based control strategies have been proposed for drug infusion systems [9]. Robust multiple-model adaptive architecture is investigated in [10], while the fractional order based PID controller is developed in [11]. In addition, a hardware-in-the-loop simulation has been applied for controlling the MAP [12]. A PressorSim software [13] was designed for simulating and monitoring hypotensive patient blood pressure response to intravenous (IV) drugs such as SNP.

Fuzzy systems are one of the adequate soft computing techniques that allow designers and/or researchers to develop a good controller for nonlinear systems based on human experience and knowledge instead of a convenient mathematical approach [14]. In the case of an undermined or ill-defined mathematical model of complex systems, fuzzy logic can be applied to define the relation between input and output. Therefore, due to the uncertainty and inevitably vague of the medical system, the fuzzy rule-based controller is a powerful and victorious technique to be applied in a medical application such as signal processing, medical image processing, diagnosis, therapy, and control as it can handle much of uncertainty exist in biological systems [15–18]. All aforementioned fuzzy controllers can be classified as a type-1 fuzzy control. However, the main issue of type-1 fuzzy controllers is their limitation in the presence of a higher degree of uncertainty in nonlinear systems [19]. They have a deficiency in describing the model because of its partial representation ability. Hence, interval type-2 fuzzy (IT2F) techniques present an extension to a type-1 fuzzy and have the ability to well-defining parameter uncertainties and the nonlinearity of the model [20]. Then, IT2F controllers can be more effective than typer-1 fuzzy counterparts in many real-world applications [21–23]. Additionally, there has been significant research work on IT2F controllers, as they can deal with the uncertainty in a class of nonlinear systems [24–26]. With the use of the lower and upper membership functions, IT2F can effectively capture the parameter uncertainties which result from different sources, such as measurement error, noise, and imprecise models. Also, IT2F systems provide some unique partitions of the input discourse, which type-1 fuzzy systems cannot offer [25]. Based on these advanced capabilities, IT2F systems have been applied in several medical applications [27–29]. Nevertheless, many of the existing systems still suffer from large setting time, instability, inaccurate or incompatibility to a wide range of physiological variations in patient conditions and patient response to IV drugs. Furthermore, medical devices are always placed near the patient to monitor and control his health signals, while the medical staff must interact with the patient directly.

Nowadays, with infectious diseases and COVID-19 outbreaks, there is a necessity to use telemedicine techniques and the remote control of medical devices to avoid the spread of infection among physicians, nursing staff, and patients. Therefore, the use of network-connected systems and telemedicine have recently emerged worldwide in healthcare and clinical procedures [30]. A networked control system (NCS) is a closed-loop system in which the feedback control loops are connected by a communication network such that the control and feedback signals of the system are transmitted among all its components through a common TCP/IP network. The design and analysis of NCSs have gained significant attention in the last decade [31,32]. Moreover, there are many advantages of applying NCS in the medical field, such as reducing the capacity of wired systems near the patient’s bedside, easy troubleshooting and maintenance of medical devices and being more reliable as compared to traditional controllers and minimizing direct contact with critically infected patients.

Metaheuristic optimization algorithms, e.g., swarm intelligence [33], present an efficient technique for auto-tuning parameters of NCS in complex medical procedures, such as the automatic management of SNP infusion rates for hypertensive patients. Teaching-learning-based optimization (TLBO) is a recent metaheuristic optimization algorithm [34]. The TLBO algorithm was proposed to simulate the effect of the influence of a teacher on learners during the learning process. It has been employed to solve many optimization problems, such as tuning PID and fuzzy controllers for different power systems [35,36]. Here, the TLBO algorithm plays an important role in automatically finding the best parameter values of the proposed controller.

The main challenges for a feedback control system are to propose a powerful controller to reduce the complexity of the proposed systems in the existence of several uncertainties, such as various patient sensitivities, outside disturbances, or noise effects. This work aims to propose a networked Interval type2 fuzzy (IT2F) controller. According to the literature studies, the IT2F controller is capable of managing uncertainties in an effective way, and the networked system will reduce the complexity of the proposed scheme. An effective control system requires tuning controller parameters to get optimal values for achieving a good closed-loop response. The TLBO algorithm is used to obtain the optimal controller’s parameters. The proposed controller is conducted to study the robustness performance for noise rejection at the control output and feedback. As a result, the objective of this work is to construct an intelligent networked control system to keep the patient’s MAP at the required level in a closed-loop manner and reduce the effect of noise and perturbation.

This study aims at developing a new networked IT2F control system for automated vasoactive drug infusion remotely. The proposed IT2F controller achieved the following advancements:

Handling hypertensive MAP of patients under different drug sensitivities and noise disturbance.

The remainder of this article is organized as follows. Section 1 presents basic definitions and background of IT2F control, networked control system, and the optimization algorithm of TLBO. Section 2 describes the mathematical patient model linked to the developed drug delivery system. Section 3 demonstrates simulation results and evaluations for all tested cases of anti-hypertensive drug infusion, comparing with the performance of traditional networked controllers under the same circumstances. Sections 4 and 5 give a discussion and conclusions of study findings, respectively, highlighting the future directions of this research work.

Professor J. M. Mendel and his coworkers have proposed the first complete type-2 fuzzy inference process, developed various type-2 fuzzy systems, and established their computational principles and foundations since the mid-1990s. In [37], a designed model is successfully used for modelling imprecision and uncertainties utilizing the IT2F system. With the interval between lower and upper membership functions, IT2F uses the footprint of uncertainty which does not exist in a type-1 fuzzy set. Structurally, IT2F contains an extra component called type-reducer which distinguishes it from type-1 fuzzy, where the form of the rules is the same for both types. Likewise, the secondary membership function of the interval type-2 fuzzy set is constant. A schematic diagram of the basic IT2F control system is shown in Fig. 1.

In the IT2F system, the fuzzy set (A~) is bounded by the upper MF (μ¯A~) and lower MF (μ_A~). For K rules, the steps of the singleton fuzzification procedure are summarized next to compute the output control signal of the IT2F logic controller [37]: 1.

Evaluate the membership interval of xi′ on each IT2 fuzzy set A~ik,[μ_A~ik(xi′),μ¯A~ik(xi′)], where (i=1,⋯,N;k=1,⋯,K) and N is a number of input variables.

(1) [f_k,f¯k]=[μ_A~1k(x1′)×⋯×μ_A~Nk(xN′),μ¯A~1k(x1′)×⋯×μ¯A~Nk(xN′)] 3.

Apply type-reducer algorithm to convert the IT2F output set into type-1 reduced fuzzy set:

(2) yll=∑i=1LLf¯ky_k+∑i=LL+1Kf_ky_k∑i=1LLf¯k+∑i=LL+1Kf_k

(3) yrr=∑i=1RRf_ky¯k+∑i=RR+1Kf¯ky¯k∑i=1RRf_k+∑i=RR+1Kf¯kwhere y¯k and y_k are the upper and lower value of the interval for the consequence term of IT2F system, y_ll and y_rr are the left and right endpoints for the type-reduced sets. The Kernik-Mendel (KM) algorithm [38,39] is applied to find the switched points (LL, RR) for y_ll and y_rr. 4.

Obtain the defuzzied output:

(4) y=yrr+yll2where y is a crisp output from the IT2F controller.

A networked control system (NCS) is a system in which the feedback control system is formed by a communication network, and all signals of the system are transmitted among all components through a common network [31]. The proposed NCS for the drug infusion system is shown in Fig. 2. It has many advantages as compared to traditional controllers, minimizing direct contact with infected patients or dangerous dynamical systems. Therefore, there are numerous potential applications in which the NCS can be applied, such as aerospace systems, power grid systems, smart cities and electric vehicles. With the spread of infectious diseases nowadays, the use of NCS in medical engineering has become an urgent necessity to avoid the spread of infection from patients to the operator. In addition, with the advanced progress of developing internet of medical things (IoMT) devices, the NCS plays an important role in modern healthcare to enable mobility, interoperability, and more flexibility for accomplishing the effectiveness of the medical care.

The TLBO algorithm is developed by Rao et al. [34] and it is based on the teacher’s influence on students’ performance. In this optimization method, the population of the optimization problem is described by a set of students, the class subjects offered to students represent the design variables, and the outcome of the student is equivalent to the ‘fitness’ value of the optimization problem. The teacher presents the best solution for the entire population. In addition, the design variables are the parameters involved in the objective function of the given optimization problem, and the best answer is the best value of this objective function. The TLBO algorithm is split into two sections, which are the Teacher phase and the Student phase, as shown in Fig. 3. The TLBO is a simple optimization technique and does need many initialized parameters to be adjusted compared to other metaheuristic optimization techniques. TLBO uses the mean value (Mean) of the population, as given (5), where Xnew and Xold present the new and existing solutions, respectively. Xteacher is the best solution. TF is a teaching factor, and r is a random number between zero and 1.

(5) Xnew=Xold+r(Xteacher–(TF)Mean)

To achieve a suitable closed-loop response, it is necessary to adjust the controller parameters to obtain optimal values. The TLBO algorithm is utilized for obtaining the optimal parameters of the controller. The study investigates the performance of the proposed controller in terms of noise rejection at the control output and feedback to determine its robustness. Thus, our goal is to create an intelligent networked control system that maintains the patient’s MAP at the required level in a closed-loop fashion while minimizing the impact of noise and disturbance. The TLBO algorithm is used to find the best parameter values of the proposed networked Interval Type 2 fuzzy (IT2F) controller for an anti-hypertensive drug delivery system. An objective function f_obj of the optimization problem is defined as the integral time absolute error (ITAE), which is the summation of the absolute error weighted by the elapsed time. The proposed controller scheme is constructed to minimizing the disparity between the target blood pressure value and the actual measurement of the patient’s blood pressure with quick responsiveness under the presence of external disturbances and variation of patient parameters. To fulfill this goal, the gains of the IT2F controller (K_e, K_de, K_p, K_I, and K_W) have been fine-tuned using the TLBO algorithm to get the best values of the gains that achieve the required response.

In this paper, a continuous third-order single-input-single-output (SISO) transfer function with time delay is addressed to conduct the response of change in patient’s MAP to infusion rates of SNP drug [38], as expressed by

(6) GPR(s)=ΔPMAP(s)USNP(s)=Ksen(1+Td3s)e−sτd(1+Td1s)[(1+Td3s)(1+Td2s)−αp]where ΔPMAP is the change of patient’s MAP due to the effect of SNP drug, USNP the input signal (drug infusion rate); τd pure delay constant; Td1,Td2 and Td3 time constants of a vasoactive drug infusion system; αp is recirculation fraction of drug; Ksen is a sensitivity parameter of the patient to drug.

So, the mean arterial pressure (MAP) for a patient under a vasoactive drug Infusion System is described as

(7) PMAP=ΔPMAP+Piniwhere Pini is the patient’s initial MAP in the absence of drug intervention, which is fixed at 150 mmHg [2], Fig. 4 shows the schematic diagram of the patient’s MAP model (open loop).

According to the parameters of the patient model in [2], the system’s time constants are given to be known and time-invariant (T_d1 = 50 s, T_d2 = 30 s, T_d3 = 10 s) and the time-delay parameterτ_d = 60 s. For simulations purpose, the sensitivity factor Ksen of the patient to SNP drug is assumed to be equal to −0.25 mmHg/ml/h. The main objective of the controller is to regulate the patient’s MAP and attain a value of 100 mmHg from the initial value of 150 mmHg. Nevertheless, the performance criteria for Vasoactive Drug Infusion System, given in Table 1, must be achieved in synthesizing the closed-loop control system. MAP regulation performance has some criteria that must be achieved as follow: MAP for a controlled system must not be below 70 mmHg, settling time must be less than 600 s, the steady-state error must be within ±5 mmHg of the desired level (100 mmHg) and overshoot is not more than 10 mmHg. The adjustment and control of SNP dose by a human operator is imprecise and can lead to possibly fatal results such as an overdose of a drug (more than 180 mlh⁻¹) being toxic to the patient. Also, manual adjustment is tedious, time-consuming, and may cause a poor response. Hence, it is required to design an adequate feedback controller that will overcome the problems of manual control and maintain MAP at the desired level effectively and accurately.

In this section, the proposed networked (IT2F) controller for a drug infusion system is developed. The target of the proposed technique is to remotely control the vasoactive drug infusion system to maintain the patient’s MAP at 100 mmHg. The block diagram that describes the operation of the networked control drug system to regulate MAP is given in Fig. 5. The blood pressure sensor and actuator, i.e., a drug infusion pump, are connected to the controller through a communication network that activates the drug infusion pump remotely. The desired value of blood pressure is established in the controller by the operator or physician, which is far away from the patient. Therefore, this technique will reduce direct contact with the patient and prevent the spread of infection from/to patients.

The controller scheme aims to be robust and get a fast response with a low error between the desired value and the measured blood pressure of the patient. For this purpose, the IT2F controller is proposed and implemented in a networked system, as shown in Fig. 6. The designed gains of the IT2F controller (K_e, K_de, K_p, K_I, and K_W) have been tuned by the TLBO algorithm, as described above. The proposed controller has two inputs, error (e) and rate of change of error (∆e), with the use of three triangular membership functions, which are given in Fig. 7, for error and change of error, respectively. Table 2 illustrates the FLC rules governing the relationship between the inputs and the infusion rate of the injected drug to the patient. The output of IT2F has five singleton membership functions, and the surface of the IT2F system is shown in Figs. 7 and 8, respectively. This control action, the output of IT2F controller Uaction, is sent via a network to the SNP drug pump to regulate the patient’s MAP.

For the case of fixed or normal patient sensitivity (Ksen = −0.25 mmHg/mlh⁻¹), Fig. 9 shows the response of the proposed IT2F controller for regulating high MAP compared with PID and type-1 fuzzy controllers. Also, the corresponding drug infusion rates produced by three tested controllers are shown in Fig. 9. Table 3 illustrates the characteristics of each tested controller in this study, including optimized control parameters, settling time, maximum overshot or undershot of regulated blood pressure, and steady-state error. Classical PID control caused a relatively long settling time of 515 s with a small hypotension of 7.20 mmHg because a high drug infusion rate is released at about 126.6 mlh⁻¹. In the case of the fuzzy type-1 controller, no hypotension is produced in an intermediate settling time of 319 s, and the drug infusion of 101.25 mlh⁻¹ is achieved. However, this controller could not reach the targeted MAP and has a relatively high value of the steady-state error of 0.63 mmHg. In contrast, the proposed networked IT2F control showed a good performance by achieving a minimal settling time of 221 s and a very small hypotension of 0.85 mmHg is resulted. The steady-state error and maximum drug rate are 0.08 mmHg and 127.2 mlh⁻¹, respectively, within accepted medical ranges.

External noise is added here to the measured MAP to simulate practical scenarios of a clinical environment and to test the stability of all networked controllers. Therefore, a white noise signal with a variance of 1 mmHg is added to the proposed drug delivery system, which results in peak-to-peak 7.0 mmHg. For the case of normal patient sensitivity with the predefined noise signal, the MAP output and drug infusion rate for all tested controllers are shown in Fig. 10. Optimized PID control can regulate MAP successfully within accepted medical ranges, but it takes a long settling time of 370 s and the steady error of approximately 0.8 mmHg. Fuzzy type-1 showed a noisy drug infusion rate with a variance of 20 mlh⁻¹ and the highest value of the steady-state error of 1.4 mmHg among other controllers, as illustrated in Table 4. The proposed IT2F controller is still the best option to handle measured noisy MAP with a minimum settling time of 345 s and the lowest value of the steady-state error of approximately 0.5 mmHg, while it infuses high maximum drug infusion of 146 mlh⁻¹ under the maximum accepted range of 180 mlh⁻¹.