Design and Implementation of a Multi-Loop Type 2 Fuzzy PID Control for an Inverted Two-Input-Two-Output System

Abstract

This paper addresses a multi-loop interval Type-2 fuzzy logic proportional-integral-derivative (PID) control for a benchmark first order plus dead time system (FOPDT). In practice, Type-1 FLCs use crisp membership functions, which leads them less effective in modelling uncertainty in real-world systems where data and expert knowledge may be imprecise or noisy. In addition, this also Type-1 FLCs cannot explicitly handle uncertainties in the system dynamics, measurement noise, or modelling errors. This limits their robustness in highly dynamic or uncertain environments. Therefore, to mitigate these above mentioned draw backs of Type 1 FLC, Type 2 FLC is widely preferred. Basically, in Type 2 FLC normally incorporated an additional layer of uncertainty within the membership functions, to make more suitable for uncertain environments. In this work, the Takagi-Sugeno Fuzzy Inference System (FIS) is employed to construct the fuzzy logic system (FLS). This study comprehensively analyzes type-1 and type-2 FLS performances for a benchmark multivariable system. The simulation results reveal that the type-2 FLS outperforms the type-1 FLS and conventional PID controller. Also, to exhibit a concrete analysis, a performance analysis table has been incorporated by considering different performance indices to provide a clear visualization of output performances. The stability analysis of the system has also been described by considering the frequency domain analysis.

Keywords

multi-loop system fuzzy logic systems decoupling decentralized control

1 Introduction

In current era, most systems in industrial and chemical processes have several inputs and outputs with intricate inter- actions among them. These higher-order systems complicated dynamics and interrelated variables might make managing and regulating them difficult. Typically, the process control plants are tedious due to the integration of different types of equipment and variables. Designing the controller is crucial for this process due to the interactive and non-linear system behaviour. Advanced control strategies are needed in such systems to guarantee stability and optimal performance. Although several controllers have been developed, but still proportional-integral-derivative (PID) controllers are widely preferred owing to its easy implementation and simple structure. Conventional PID controllers are obviously the most often used controllers in industry because of their easy implementation and straightforward design. The tuning of PID controllers has been reported in literature using a great range of design strategies (Chai et al., 2024; Dey et al., 2024; Ningsih et al., 2024). Among them are Ziegler and Nichols, Cohen and Coon, pole placement design techniques, internal model control, and many more approaches. PID controllers may be a useful tool for linear system control; yet, they may not be able to provide a good closed loop (Cantera-Cantera et al., 2025; Cervantes et al., 2023; Duddeti & Naskar, 2025; Liu et al., 2025; Nie et al., 2022; Utami et al., 2022) control performance in nonlinear systems or with unknown process models. In addition to this also the conventional PID controller faces problems particularly to variation of operating conditions, that necessitates the development of superior controller approaches. Therefore, to tackle this type-1 fuzzy logic control is widely adopted to control the nonlinear systems. From the literature it is witnessed that type-1 fuzzy PID controllers are provided adequate loop performances rather than conventional PID controllers. From the past few decades, a significant research work on type-1 fuzzy PID controller have been successfully deployed in different engineering fields (Al-Fandi et al., 2012; Castillo et al., 2016; Mendel et al., 2014; Prusty et al., 2015). The type-1 Fuzzy Logic Controller (FLC) has been widely implemented in numerous real-world applications (Al-Fandi et al., 2012; Castillo et al., 2016; Mittal et al., 2020). Nevertheless, a significant limitation of this controller is its inability to effectively manage the inherent uncertainties in the membership functions (MF) of both input as well as output variables. In (Wu & Mendel, 2010), the fuzzy controller is integrated with the PID controller and then the combined fuzzy-PID controller is applied to the tank level system. This paper provides the comparison of transient response and error indices of PID, fuzzy and fuzzy-PID controllers respectively. The fuzzy-PID controller has provided superior performances. Experts’ collective expertise and knowledge typically establish fuzzy rules and MFs. Nevertheless, it is essential that once the MFs have been selected for controller design, it is impossible to accurately represent the uncertainties associated with the actual degree of membership functions. Fuzzy rules can cause delay in three ways: (a) The terms antecedent and consequent may be interpreted differently by different individuals (Castillo et al., 2016) (b) Expert groups might provide varying outcomes for the same rule, (c) Uncertainty in the antecedent and consequent can be propagated into MFs due to noisy training data (Prusty et al., 2015; Zadeh, 1975). Thus, type-1 fuzzy sets are the MFs of type-1 FLC, cannot manage rule uncertainty. Therefore, type-1 FLC efficiency may suffer, especially when the plant is disturbed (Sakalli et al., 2021).

From the perspective of input-output correlation, the type-1 fuzzy PID controller structures resemble those of the traditional PID controllers. It is witnessed that type-1 fuzzy PID controller have been validated successfully in various complex and nonlinear system including robotics, unmanned aerial vehicle, renewable energy, chemical industries etc. Despite widespread application of type-1 fuzzy logic proportional-integral (PI) controller, still it faces difficulties while minimizing the effect of uncertainties in the plant. Subsequently type-1 fuzzy sets might not necessarily provide the best optimal performance. Therefore, from the literature it is noticed that type-1 FLC is fails to deliver adequate loop performances when the system is involved with a higher degree of uncertainty. Thus, to overcome this drawback of the type-1 FLC, type-2 FLC has been developed. Moreover, to tackle the challenges of uncertainty in MFs and fuzzy rules, researchers have introduced interval type-2 fuzzy logic control (IT2FLC), acknowledging the limitations of type-1 FLCs. Initially, Lotfi Zadeh addressed the type-2 fuzzy logic controller (T2FLC) in (Mendel, 2014). Type-2 FLSs utilize a three-dimensional fuzzy model that incorporates a Footprint of Uncertainty (FOU) to define their MFs (Dereli et al., 2011). Several type-2 FL PI controller architectures have been documented in the literature, most of which were developed using heuristics. Type-2 fuzzy systems differ from type-1 systems primarily in their use of MFs, where the membership values themselves are fuzzy rather than precise (Castro et al., 2007).

Type-1 assumes that membership can be assigned a precise numeric value, which is not always the case. The design of type-1 FLSs is significantly influenced by the shape of the MF. Zadeh has developed type-2 fuzzy sets (T2FSs) to incorporate uncertainty into fuzzy systems. These are enhancements of type-1 fuzzy sets (T1FSs) that introduce an additional degree of freedom to account for ambiguity within systems. In a type-1 FLC (T1FLC), the rules are typically derived from imprecise expert knowledge, as the insights provided by different experts may vary or lack consistency (Dereli et al., 2011; Sakalli et al., 2021). T2FLC is a novel approach that overcomes the restrictions of T1FSs and provides unique features in order to accommodate the uncertainty associated with expert knowledge (Du & Ying, 2010; Wu & Tan, 2010). The primary challenge with T2FLS lies in its computational complexity, which is significantly higher than that of T1FLS. To address this, (Aliasghary et al., 2012) introduces a simplified analytical variant known as the interval type-2 fuzzy logic system (IT2FLS). In this system, the output of the type-2 fuzzy interval is represented as an uncertain interval. Recently, the use of IT2FLSs has gained significant attention in various applications because of their effectiveness in handling uncertainties (Castillo & Melin, 2014; El-Bardini & El-Nagar, 2014; Kumbasar & Hagras, 2015; Raj & Mohan, 2020; Sakalli et al., 2021; Shi, 2022; Zhou et al., 2021). These efforts focus on controllers referred to as” black box” controllers, where the input-output mathematical relationships are not explicitly known. The internal structure of the IT2FLC closely resembles that of its type-1 counterpart. The key distinction, however, lies in the rule base, where at least one fuzzy set (FS) is an interval type-2 fuzzy set (IT2FS). Therefore, before performing a defuzzification procedure, a type-reducer must convert them into a T1FS (Wu, 2012).

Often, IT2FLS exhibits adequate performances due to the FOU in their MFs that provides more resilient performances. From the literature (Karnik & Mendel, 2001; Nishanth et al., 2024; Sharma & Kumar, 2022), it can be noticed that how IT2FLCs are more robust and smoother than T1FLC. Nevertheless, because of their intricate internal structure, T2FLCs are challenging to analyze and build. The author in (Kumbasar & Hagras, 2015) also noted that the creation of an IT2FLC involves numerous choices, such as the construction of the rule base and the selection of the type and quantity of IT2FS to use. Several studies have designed and explored the input-output relationships of IT2FLCs in this context. Consequently, IT2FLCs have gained significant attention in research, particularly in control applications, due to their enhanced ability to manage uncertainties and nonlinearities. Thus, Researchers have successfully applied and managed various systems, including autonomous mobile robots, plant control, bioreactor management, and pH regulation. Even though extensive application of T2FLC has been explored in numerous area; however still there is no significant application has been noticed in the field of process control with delay system. Therefore, in this paper type-2 fuzzy PID is adopted for a two input two output (TITO) system. Moreover, a depth study is explored to compare between T1FLC and T2FLC. It employs the use of triangular and trapezoidal MFs with three and five linguistic variables. (Figure 1)

Figure 1.

Developments of Fuzzy Logic Controllers (From Conventional to Advanced Variants) (Sain et al., 2025).

Our current work focuses on the implementation of type-2 FLS PID controller for an inverted TITO system. The comparisons between the responses of type-1 and type-2 FLS PID controller for an inverted TITO system are discussed through simulation. We conduct analysis by comparing triangular and trapezoidal membership function using type-1 and type-2 fuzzy logic controller for plant 1 and plant 2. Various performance indices such as integral absolute error (IAE), integral squared error (ISE), integral time absolute error (ITAE) and integral time squared error (ITSE) are calculated for inverted TITO system. This work also includes a stability analysis using frequency response methods, such as Bode plots and Nyquist plots.

The rest of the paper is arranged as follows. Section 2 provides basic preliminaries of fuzzy logic control. The type-2 fuzzy PID controller along with a systematic in depth comparison with type-1 fuzzy PID controller has been described in Section 3. Various simulation responses of type-1 and type-2 fuzzy PID controller and stability analysis have presented in Section 4. Finally, conclusions are made in section 5.

2 Problem Statement

In general, multivariable liquid level system involves numerous control challenges due to its inherent nonlinearity, strong coupling effects between input and output variables, and time-delay characteristics. Thus, particularly in this process, loop interactions possess difficulties in achieving precise level control. In addition to this, also external disturbances, parameter variations, and uncertainties further imposes typical challenges in system performances. In fact, traditional control methods often struggle with the slow response and sensitivity to fluctuations issues. Therefore, to overcome these challenges in this work, an interval type 2 fuzzy inference system is designed whose rule base, membership functions, and scaling factors are optimized to ensure that the control objectives of the plant such as good servo and regulatory responses, fast settling time, and adequate disturbance rejection are achieved under various operating conditions. (Figure 2)

Figure 2.

Schematic Representation of the Type-2 FLS PID for MIMO System.

3 Type-2 Fuzzy Logic PID Controller Design

In this work, the control output is determined by using the two inputs: the error and the derivative of the error. The two inputs can be normalized by using input scaling factors SF_e and SF_de. The output scaling factors (SF₁ and SF₂) can be used for converting normalized to general control output. To pursue the controller design here two inverted TITO plant is considered.

3.1 Choice of Scaling Factors

The inverted TITO system has two plants described by the transfer function as $P_{1} = \frac{0.3924 e^{- 3.831 s}}{620.765 s + 1}$ and $P_{2} = \frac{0.5 e^{- 19.46 s}}{632.74 s + 1}$ , respectively. For the plant P₁, the input scaling factors are as follows:

\begin{aligned} S F_{e} & = \frac{1}{r (t) - y (t)} \end{aligned}

(1)

\begin{aligned} S F_{d e} & = 1.916 S F_{e} \end{aligned}

(2)

The output scaling factors of plant P₁ are as follows:

\begin{aligned} S F_{1} & = \frac{1}{0.83374 S F_{e}} \end{aligned}

(3)

\begin{aligned} S F_{2} & = 620.765 S F_{1} \end{aligned}

(4)

where r(t) and y(t) are the reference and system output values.

Similarly, for plant P₂, the input scaling factors are as follows:

\begin{aligned} S F_{d e} = 9.73 S F_{e} \end{aligned}

(5)

The output scaling factors of plant P₂ are as follows:

\begin{aligned} S F_{1} & = \frac{1}{4.965 S F_{e}} \end{aligned}

(6)

\begin{aligned} S F_{2} & = 632.74 S F_{1} \end{aligned}

(7)

3.2 Fuzzifier

The fuzzification is the first process to convert a crisp value into type-2 fuzzy value. The input variables error and derivative of error are fuzzified by three linguistic variables (negative, zero and positive). Similarly, the input variables error and derivative of error are fuzzified by five linguistic variables such as negative big, negative small, zero, positive small and positive big, respectively. The fuzzy sets are considered as triangular and trapezoidal membership function. The fuzzified input variables for type-1 and type-2 triangular MF are shown in Figure 3(a) and Figure 3(b). Similarly, the fuzzified input variables for type-1 and type-2 trapezoidal MF are shown in Figure 4(a) and Figure 4(b), respectively.

Figure 3.

Triangular MF of 3 Linguistic Variables: (a) Type-1 and (b) Type-2.

Figure 4.

Trapezoidal MF of 3 Linguistic Variables: (a) Type-1 and (b) Type-2.

3.3 Rule Base

The fuzzy logic rule base is developed based on the control action of the given TITO system. The Takagi Sugeno fuzzy inference system is employed in this work. As there are 3 linguistic variables in each input variables (error and change in error), there are 9 rule base to control the FIS. The control rule base of the given FIS is shown in Table 1.

3.4 Type Reducer

Type reducer is one which converts type-2 fuzzy output of an inference engine into type-1 fuzzy output, allowing standard defuzzification methods to be applied. Type reduction acts as an intermediate step between the fuzzy inference stage and defuzzification, which produces a crisp output. In a T2FLS, each element in the fuzzy set has a range of possible membership values rather than a single fixed value. This range is represented by a secondary membership function, resulting an IT2FS. Type reduction aims to collapse this range by summarizing the fuzzy set with two boundary values, creating a simpler interval that still captures the essence of the uncertainty. The Karnik-Mendel (KM) algorithm is used in this plant for reducing type-2 fuzzy output to type-1 fuzzy output.

Karnik-Mendel Algorithm

Karnik-Mendal Algorithm is widely used to find the center of type-2 fuzzy outputs. It presents iterative procedures to calculate $y_{r}, y_{l}$ (Karnik & Mendel, 2001).

(1)
Assume that $y_{r}^{1} \leq y_{r}^{2} \leq \dots \leq y_{r}^{N}$ .
(2)
Calculate $y_{r}$ as $y_{r} = \frac{\sum_{i = 1}^{N} f_{r}^{i} y_{r}^{i}}{\sum_{i = 1}^{N} f_{r}^{i}}$ by initially setting $f_{r}^{i} = \frac{\bar{f^{i}} + \underline{f^{i}}}{2}$ and let ${y_{r}}^{'} ≅ y_{r}$ .
(3)
Find $S ∋ y_{r}^{S} \leq y_{r} \leq y_{r}^{S + 1}$ where $S \in [1, N - 1]$ .
(4)
Calculate $y_{r} = \frac{\sum_{i = 1}^{N} f_{r}^{i} y_{r}^{i}}{\sum_{i = 1}^{N} f_{r}^{i}}$ with $f_{r}^{i} = \underline{f^{i}}$ for $i \leq S$ and $f_{r}^{i} = \bar{f^{i}}$ for $i > S$ and let $y_{r}^{^{″}} = y_{r}$ .
(5)
If ${y_{r}}^{″} \neq {y_{r}}^{'}$ then go to step 6 otherwise stop and set $y_{r}^{^{″}} = y_{r}$
(6)
Set equal to $y_{r}^{^{″}}$ and return to step 3.

Table 1.
Fuzzy Rules for 3 Linguistic Variables.

Change in Error

CS N Z P

Error N N N Z

Z N Z P

P Z P P

The procedure is to compute $y_{l}$ which is similar to compute $y_{r}$ by replacing the variables.

Remark-1

In this work, the optimum settings of the Fuzzy Logic Controllers (FLCs) were determined without using any formal optimization methods. Here, a manual tuning approach was adopted, guided by expert knowledge and iterative simulation. Initially, the membership functions and rule base were designed based on the known dynamics of the system and previous experience. These settings were then refined through trial and error by observing system performance metrics such as rise time, overshoot, settling time, and steady-state error. Adjustments were made iteratively until the desired control objectives were achieved
3.5 Defuzzifier

		Change in Error
Error	N	N	N	Z
Z	N	Z	P
P	Z	P	P

A defuzzifier is the component that converts fuzzy output values, generated by the fuzzy inference system, into a single crisp output. This step is essential because fuzzy logic deals with “degrees of truth” rather than binary true or false values, and the defuzzifier translates the fuzzy sets (ranges of possible output values) into a precise, actionable value. In order to make computationally efficient, the Sugeno fuzzy inference system is applied in the given plant. In this system, the output is the linear combination of the input variables. It uses a weighted average for defuzzification. The overall output is computed as in (8).

\begin{aligned} U = \frac{\sum_{i = 1}^{n} p_{i} f_{i}}{\sum_{i = 1}^{n} p_{i}} \end{aligned}

(8)

where p_i is the firing strength of i^th rule f_i is the output of the i^th rule.

3.6 Control Surface of Type-1 and Type-2 Fuzzy Inference Engine

The control surface is defined as a visual representation of the behavior of the system. The purpose of control surface shows that the different combination of input values which affects the output. The control surface of both type-1 and type-2 fuzzy logic controller relation between the error, change in error and control signal of triangular membership function are described in Figure 5. The control surface of both type-1 and type-2 fuzzy logic controller relation between the error, change in error and control signal of trapezoidal membership function are presented in Figure 6.

Figure 5.

Triangular MF Control Surface of 3 Linguistic Variables: (a) Type-1 and (b) Type-2.

Figure 6.

Trapezoidal MF Control Surface of 3 Linguistic Variables: (a) Type-1 and (b) Type-2.

4 Results and Discussion

The proposed controller is compared with T1FLC is applied in FOPDT with inverted TITO system is discussed. The first order plus dead time system has been described by the plants P₁ and P₂ respectively.

4.1 Step Response

The step response for plant $P_{1}$ is shown in Figure 7(a) and the corresponding controller output is shown in Figure 7(b). The performance parameter (rise time) and performance indices such as IAE, ISE, ITAE and ITSE are calculated and presented in Table 2. It has been observed from Table 2 that the type-2 fuzzy PID controller exhibits superior performance as compared to its type-1 counterparts, primarily due to its reduced error rate compared to the type-1 fuzzy PID controller in Plant 1.

Figure 7.

For Tank 1: (a) Step Response and (b) Controller Output.

Table 2.

Performance Indices for Plant 1.

Performance	Conventional	T1FLC	T1FLC	IT2FLC	IT2FLC
Indices	PID	(Triangular MF)	(Trapezoidal MF)	(Triangular MF)	(Trapezoidal MF)
Rise Time	50.762	3.4068	3.4086	3.4131	3.4169
Settling Time	365.1281	22.6334	32.6112	18.8094	18.527
IAE	48.7	6.6996	7.0999	6.3902	6.4213
ITAE	5034	37.375	52.6288	31.1392	31.8982
ISE	19.2804	5.2859	5.2997	5.2605	5.2618
ITSE	525.3048	20.1536	20.4965	19.7433	19.7682

Figure 8.

For Tank 2: (a) Step Response and (b) Controller Output.

Similarly, the step response for plant $P_{2}$ is shown in Figure 8(a) and the corresponding controller output is shown in Figure 8(b). The performance parameter (rise time) and performance indices such as IAE, ISE, ITAE and ITSE are calculated and presented in Table 3. It has been observed from Table 3 that the type-2 fuzzy PID controller exhibits superior performance as compared to its type-1 counterparts, primarily due to its reduced error rate compared to the type-1 fuzzy PID controller in Plant 2.

4.2 Disturbance Rejection

For disturbance rejection, the external disturbance is added in the normal step response by using step with amplitude 0.2 at time 450 s for both plants $P_{1}$ and $P_{2}$ . The disturbance rejection for the plant $P_{1}$ is shown in Figure 9(a) and the corresponding controller output is shown in Figure 9(b). The performance parameter (rise time) and performance indices such as IAE, ISE, ITAE and ITSE are calculated and presented in Table 4.

Table 3.
Performance Indices for Plant 2.

Performance Conventional T1FLC T1FLC IT2FLC IT2FLC

Indices PID (Triangular MF) (Trapezoidal MF) (Triangular MF) (Trapezoidal MF)

Rise Time 29.7153 15.888 15.8876 15.888 15.888

Settling Time 348.0623 130.5862 183.1224 87.2168 87.3097

IAE 63.3339 34.3176 37.275 32.177 32.5801

ITAE 5816.9 879.8508 1476 654.4422 703.7558

ISE 34.0718 26.4297 26.5487 26.2762 26.2816

ITSE 893.0079 400.7764 415.7569 388.0103 388.8474

Performance	Conventional	T1FLC	T1FLC	IT2FLC	IT2FLC
Rise Time	29.7153	15.888	15.8876	15.888	15.888
Settling Time	348.0623	130.5862	183.1224	87.2168	87.3097
IAE	63.3339	34.3176	37.275	32.177	32.5801
ITAE	5816.9	879.8508	1476	654.4422	703.7558
ISE	34.0718	26.4297	26.5487	26.2762	26.2816
ITSE	893.0079	400.7764	415.7569	388.0103	388.8474

Figure 9.

Disturbance Rejection for Tank 1: (a) Step Response and (b) Controller Output.

Table 4.

Performance Indices for Plant 1: Disturbance Rejection.

Performance	Conventional	T1FLC	T1FLC	IT2FLC	IT2FLC
Indices	PID	(Triangular MF)	(Trapezoidal MF)	(Triangular MF)	(Trapezoidal MF)
Rise Time	50.3924	3.4073	3.409	3.4139	3.4174
Settling Time	512.0924	513.316	482.656	462.8494	461.7584
IAE	58.4277	8.1855	8.7797	8.1965	8.1018
ITAE	10281	723.7102	831.2848	868.8082	807.6121
ISE	20.1466	5.4978	5.5121	5.5013	5.497
ITSE	933.2494	116.1243	116.7526	128.878	126.3829

It has been observed from Table 4 that the type-2 fuzzy PID controller exhibits superior performance as compared to its type-1 counterparts, primarily due to its reduced settling time and eliminates oscillations compared to the type-1 fuzzy PID controller in Plant 1.

Similarly, the step response for plant $P_{2}$ is shown in Figure 10(a) and the corresponding controller output is shown in Figure 10(b). The performance parameter (rise time) and performance indices such as IAE, ISE, ITAE and ITSE are calculated and presented in Table 5. It has been observed from Table 5 that the type-2 fuzzy PID controller exhibits superior performance as compared to its type-1 counterparts, primarily due to its reduced settling time and eliminates oscillations compared to the type-1 fuzzy PID controller in Plant 2.

Figure 10.

Disturbance Rejection for Tank 2: (a) Step Response and (b) Controller Output.

Table 5.

Performance Indices for Plant 2: Disturbance Rejection.

Performance	Conventional	T1FLC	T1FLC	IT2FLC	IT2FLC
Indices	PID	(Triangular MF)	(Trapezoidal MF)	(Triangular MF)	(Trapezoidal MF)
Rise Time	29.4104	15.902	15.889	15.9081	15.9029
Settling Time	564.5043	527.47	550.8634	504.0505	500.7418
IAE	74.0038	41.9192	45.7359	41.0414	40.8578
ITAE	11208	4532.9	5723.5	4924.6	4676.5
ISE	35.6198	27.5101	27.6481	27.4914	27.4647
ITSE	1617.6	902.4027	928.5325	954.4805	939.7272

Figure 11.

Magnitude and Phase Diagram of (a) Plant 1 (b) Plant 2.

4.3 Stability Analysis of Plant

Stability analysis of plant is an important study to determine whether a system will remain in a desirable state or performance level over time without diverging. A stable plant maintains its output within a bounded range when subjected to an input or disturbance. Stability analysis is essential for designing controllers and ensuring safety, reliability, and efficiency. The stability of the plant 1 and plant 2 is analysed by using frequency response techniques such as bode plot and Nyquist plot. The bode diagrams of plant 1 and plant 2 are shown in Figure 11(a) and Figure 11(b). The stability of the plant is analysed from gain margin (GM) and phase margin (PM). The GM of plant 1 is 56.2 at phase crossover frequency of 0.41 rad/sec and PM is infinite. Hence, the plant 1 is stable as both GM and PM are positive. Similarly, plant 2 is also stable as GM is 40.3 at phase crossover frequency of 0.0817 rad/sec and PM is infinite.

The stability of the plant is also analysed using Nyquist plot by Nyquist stability criterion. The Nyquist stability criterion helps to determine closed-loop stability based on the open-loop frequency response. It is observed from Figure 12(a) that there is no encirclement of −1 + j0 point of Nyquist plot. Hence, the closed loop comprises of plant 1 is stable. Similarly, the closed loop comprises of plant 2 is also stable as there is no encirclement of −1 + j0 point which is shown in Figure 12(b). The plant 1 and plant 2 are both stable by using bode plot as well as Nyquist plot.

Figure 12.

Nyquist Diagram of (a) Plant 1 (b) Plant 2.

5 Conclusion

In this paper, an interval type-2 fuzzy PID controller is addressed for multi-loop system. Moreover, a depth comparative analysis has been made between interval type-2 fuzzy logic PID control, type-1 fuzzy logic PID control with a conventional PID control approach. Additionally, a concrete analysis is explored to exhibits the effects of using three linguistic variables versus five linguistic variables and the impact of using triangular membership functions versus trapezoidal membership functions on system performance. From the simulation it is noticed that the IT2FLC, utilizing a triangular membership function, achieves higher accuracy and superior performance compared to T1FLC and conventional PID controller. Further, the stability analysis of the system has been demonstrated by calculating the frequency domain specifications such as GM, PM, gain crossover frequency and phase crossover frequency.

Footnotes

ORCID iDs

Soumya Ranjan Mahapatro

Nitish Katal

Sankata Bhanjan Prusty

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of Conflicting Interests

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

References

Al-Fandi

Jaradat

M. A.

Sardahi

(2012). Optimal pid-fuzzy logic controller for type 1 diabetic patients. In 2012 8th International Symposium on Mechatronics and its Applications, pp. 1–7, IEEE.

Aliasghary

Eksin

Güzelkaya

Kumbasar

(2012). Design of an interval type-2˙ fuzzy logic controller based on conventional pi controller. In 2012 20th Mediterranean Conference on Control & Automation (MED), pp. 627–632, IEEE.

Cantera-Cantera

L. A.

Maya-Rodríguez

M. C.

Palomino-Resendiz

S. I.

Lozano-Hernández

Luna

(2025). Level and flow systems identification of an industrial processes module by LSOD method for PID controllers design. Results in Engineering, 25, 104347. https://doi.org/10.1016/j.rineng.2025.104347

Castillo

Amador-Angulo

Castro

J. R.

Garcia-Valdez

(2016). A comparative study of type-1 fuzzy logic systems, interval type-2 fuzzy logic systems and generalized type-2 fuzzy logic systems in control problems. Information Sciences, 354, 257–274. https://doi.org/10.1016/j.ins.2016.03.026

Castillo

Melin

(2014). A review on interval type-2 fuzzy logic applications in intelligent control. Information Sciences, 279, 615–631. https://doi.org/10.1016/j.ins.2014.04.015

Castro

J. R.

Castillo

Melin

(2007). An interval type-2 fuzzy logic toolbox for control applications. In 2007 IEEE international fuzzy systems conference, pp. 1–6, IEEE.

Cervantes

Caraveo

Castillo

(2023). Performance comparison between type-1 and type-2 fuzzy logic control applied to insulin pump injection in real time for patients with diabetes. Mathematics, 11(3), 730. https://doi.org/10.3390/math11030730

Chai

Zhou

Cheng

Jia

Song

(2024, November). Industrial metaverse-based intelligent PID optimal tuning system for complex industrial processes. IEEE Transactions on Cybernetics, 54(11), 6458–6470. https://doi.org/10.1109/TCYB.2024.3386669

Dereli

Baykasoglu

Altun

Durmusoglu

Türksen

I. B.

(2011). Industrial applications of type-2 fuzzy sets and systems: A concise review. Computers in Industry, 62(2), 125–137. https://doi.org/10.1016/j.compind.2010.10.006

10.

Dey

Banerjee

Dey

(2024, September). Frequency-Domain tuning of a robust optimal 2-DOF fractional order PID controller for a maglev system. IEEE Transactions on Industrial Informatics, 20(9), 11348–11361. https://doi.org/10.1109/TII.2024.3403249

11.

Ying

(2010). Derivation and analysis of the analytical structures of the interval type-2 fuzzy-pi and pd controllers. IEEE Transactions on Fuzzy Systems, 18(4), 802–814. https://doi.org/10.1109/TFUZZ.2010.2049022

12.

Duddeti

B. B.

Naskar

A. K.

(2025). Model dominance index assisted enhanced generalized pole clustering technique for order reduction and application to PID controller design. Circuits, Systems, and Signal Processing, 44, 1–34. https://doi.org/10.1007/s00034-025-02998-x

13.

El-Bardini

El-Nagar

A. M.

(2014). Interval type-2 fuzzy PID controller: Analytical structures and stability analysis. Arabian Journal for Science and Engineering, 39, 7443–7458. https://doi.org/10.1007/s13369-014-1317-y

14.

Karnik

N. N.

Mendel

J. M.

(2001). Centroid of a type-2 fuzzy set. Information Sciences, 132(1-4), 195–220. https://doi.org/10.1016/S0020-0255(01)00069-X

15.

Kumbasar

Hagras

(2015). Interval type-2 fuzzy PID controllers. In Springer handbook of computational intelligence (pp. 285–294). Springer.

16.

Liu

Qiu

Mao

(2025). The PID high-type controller based on LQR and model-assisted LESO. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 239, 74–89. https://doi.org/10.1177/09596518241263538

17.

Mendel

J. M.

(2014). General type-2 fuzzy logic systems made simple: A tutorial. IEEE Transactions on Fuzzy Systems, 22(5), 1162–1182. https://doi.org/10.1109/TFUZZ.2013.2286414

18.

Mendel

Hagras

Tan

W.-W.

Melek

W. W.

Ying

(2014). Introduction to type-2 fuzzy logic control: theory and applications. John Wiley & Sons.

19.

Mittal

Jain

Vaisla

K. S.

Castillo

Kacprzyk

(2020). A comprehensive review on type 2 fuzzy logic applications: Past, present and future. Engineering Applications of Artificial Intelligence, 95, 103916. https://doi.org/10.1016/j.engappai.2020.103916

20.

Nie

Z. Y.

Wang

Q. G.

Gao

Luo

(2022). A unifying Ziegler–Nichols tuning method based on active disturbance rejection. International Journal of Robust and Nonlinear Control, 32(18), 9525–9541. https://doi.org/10.1002/rnc.5848

21.

Ningsih

Suharti

P. H.

Ghani

(2024). Performance analysis of IMC-PID controller on PCT-14 air pressure control system. MethodsX, 13, 102966. https://doi.org/10.1016/j.mex.2024.102966

22.

Nishanth

F. P.

Dash

S. K.

Mahapatro

S. R.

(2024). Critical study of type-2 fuzzy logic control from theory to applications: A state-of-the-art comprehensive survey. e-Prime-Advances in Electrical Engineering, Electronics and Energy 10, 100771. https://doi.org/10.1016/j.prime.2024.100771

23.

Prusty

S. B.

Pati

U. C.

Mahapatra

K. K.

(2015). A novel fuzzy based adaptive control of the four tank system. In Proceedings of the 2015 Third International Conference on Computer, Communication, Control and Information Technology (C3IT), pp. 1–6, IEEE.

24.

Raj

Mohan

(2020). General structure of interval type-2 fuzzy pi/pd controller of takagi–sugeno type. Engineering Applications of Artificial Intelligence, 87, 103273. https://doi.org/10.1016/j.engappai.2019.103273

25.

Sain

Praharaj

Mohan

B. M.

Yang

J. M.

(2025). Interval type-2 fuzzy PID controllers with interval of confidence and various types of footprints of uncertainty. Information Sciences, 699, 121795. https://doi.org/10.1016/j.ins.2024.121795

26.

Sakalli

Kumbasar

Mendel

J. M.

(2021). Towards systematic design of general type-2 fuzzy logic controllers: Analysis, interpretation, and tuning. IEEE Transactions on Fuzzy Systems, 29(2), 226–239. https://doi.org/10.1109/TFUZZ.2020.3016034

27.

Sharma

Kumar

(2022). Optimal interval type-2 fuzzy logic control based closed-loop regulation of mean arterial blood pressure using the controlled drug administration. IEEE Sensors Journal, 22(7), 7195–7207. https://doi.org/10.1109/JSEN.2022.3151831

28.

Shi

(2022). A unified general type-2 fuzzy pid controller and its comparative with type-1 and interval type-2 fuzzy pid controller. Asian Journal of Control, 24(4), 1808–1824. https://doi.org/10.1002/asjc.2578

29.

Utami

A. R.

Yuniar

R. J.

Giyantara

Saputra

A. D.

(2022). Cohen-Coon PID tuning method for self-balancing robot. In 2022 International Symposium on Electronics and Smart Devices (ISESD).

30.

(2012). On the fundamental differences between interval type-2 and type-1 fuzzy logic controllers. IEEE Transactions on Fuzzy Systems, 20(5), 832–848. https://doi.org/10.1109/TFUZZ.2012.2186818

31.

Mendel

J. M.

(2010). On the continuity of type-1 and interval type-2 fuzzy logic systems. IEEE Transactions on Fuzzy Systems, 19(1), 179–192. https://doi.org/10.1109/TFUZZ.2010.2091962

32.

Tan

W. W.

(2010). Interval type-2 fuzzy pi controllers: Why they are more robust. In 2010 IEEE International Conference on Granular Computing, pp. 802–807, IEEE.

33.

Zadeh

L. A.

(1975). The concept of a linguistic variable and its application to approximate reasoning-iii. Information Sciences, 9(1), 43–80. https://doi.org/10.1016/0020-0255(75)90017-1

34.

Zhou

Zhang

Tan

Dai

Duan

(2021). Design of the footprints of uncertainty for a class of typical interval type-2 fuzzy pi and pd controllers. ISA Transactions, 108, 1–9. https://doi.org/10.1016/j.isatra.2020.08.009