An interval type-3 fuzzy PID control system design and its application in solid oxide fuel cells power plant

Abstract

Compared with type-2 fuzzy sets, the secondary membership degree of interval type-3 fuzzy sets is an interval rather than crisp value, which makes interval type-3 fuzzy sets can obtain more degree of freedoms. This article studies an interval type-3 fuzzy PID controller based on interval type-3 fuzz sets. The framework of interval type-3 fuzzy PID controller is identical with type-2 fuzzy PID controller, but it contains more adjustment controller parameters and its type reduction procedure is more complex. In this paper, type reduction of interval type-3 fuzzy sets is derived from general type-2 fuzzy sets represented by α-plane and a direct NT type reduction algorithm is applied. The control effects of interval type-3 fuzzy PID controller are firstly tested by 2 nonlinear plants, the simulation results show that interval type-3 fuzzy PID controller has better control performance indexes than PID controller, type-1 fuzzy PID controller, interval type-2 fuzzy PID controller and general type-2 fuzzy PID controller. Furthermore, the interval type-3 fuzzy PID controller will be applied in rated voltage control of solid oxide fuel cells (SOFC) power plant. The output voltage control of SOFC is quite challenging because of the strong nonlinearity, limited fuel flow, and rapid variation of the load disturbance. The simulation results demonstrate the advantages and robustness of proposed interval type-3 fuzzy PID controller.

Keywords

Type-2 fuzzy sets interval type-3 fuzzy sets interval type-3 fuzzy control system type reduction solid oxide fuel cells

1 Introduction

Compared to classical sets, type-1 fuzzy sets (T1FS) can define the uncertainty of individual users’ understanding of semantic concepts, that is, intra-individual uncertainty. However, for the same semantic concept, different persons may have different understandings, that is, inter-individual uncertainty. For example, the understanding of the concept of “high temperature” is not entirely consistent between northerners and southerners. T1FS cannot describe the uncertainty between individuals, because the membership degree of each element is a certain value. Why the membership degree of “34.5” belong to “high temperature” must be 0.9 rather than 0.85? This may lead to different person define different membership function.

Also, in real control systems, there exists many sources of uncertainties, like: 1) Linguistic uncertainties in antecedent and consequent descriptions of fuzzy rules; 2) Differences in the antecedents of the same rules obtained through expert questionnaire surveys; 3) Input signal measurement noise when activating fuzzy systems; 4) Parameter uncertainties due to the applications of noisy training data for adjustment or optimization. The uncertainties of these languages and data can lead to the uncertainties of fuzzy rules antecedents and/or antecedents, ultimately reflected in the uncertainties of corresponding membership functions. Due to the fact that both language and data uncertainties are reflected in the uncertainties of membership functions, type-2 fuzzy sets (T2FS) based on fuzzy membership function representation can actually describe both language and data uncertainties and handle the uncertainties of fuzzy rules directly.

T2FS involves 2 categories, interval type-2 fuzzy sets (IT2FS) and general type-2 fuzzy sets (GT2FS). IT2FS is a special case of GT2FS [1], whose secondary membership degree is 1. The structure of type-2 fuzzy logic systems (T2FLS) is the same as type-1 fuzzy logic systems (T1FLS), except that T2FLS contains a type reduction procedure, which convers type-2 fuzzy sets to type-1 and then get the defuzzification result. Karnik-Mendel (KM) [2] and its improved type reduction algorithms are commonly applied for IT2FS. Based on α-plane representation, GT2FS type reduction can be executed by several IT2FS type reduction algorithms [3], and Torshizi, et al. reviewed some type reduction algorithms for interval and general type-2 fuzzy sets [4]. For the secondary membership degree of GT2FS is defined by membership function, so general type-2 fuzzy logic systems (GT2FLS) has more design freedoms than interval type-2 fuzzy logic systems (IT2FLS). The applications of GT2FLS indicated that it can obtain better performances than IT2FLS, such as, fuzzy clustering [5], bearing fault detection [6], brain-machine interface [7], neural network [8], controller design [9], human-machine interfaces [10], large-scale social networks [11], fuzzy regression [12] and so on.

Also, general type-2 fuzzy systems may obtain better performances in some control systems with high uncertainties compared with tradition type-1 fuzzy systems and interval type-2 fuzzy systems. The applications of general type-2 fuzzy controller has been applied in mainly fields, like mobile robot [13, 14], water tank, temperature, mobile robot and beam and ball [15], traffic signal scheduling [16], inverted pendulum plant, robotic manipulator with time-varying payloads and 3-PSP parallel robot [17, 18], 5-agents system, a multi-agent with unknown dynamics and unknown time-varying topology [19], two-wheeled self-balancing robot [20], trajectory tracking of wheeled mobile robots [21], nonlinear power systems [22], aerospace [23], airplane flight [24], steam temperature at collector outlet of trough solar thermal power generation system [25, 26], power-line inspection robots [27].

With the developments of fuzzy logic theories, higher order fuzzy logic systems have been concerned, called type-n fuzzy logic systems [28]. Although, the ability of handing uncertainties of type-n fuzzy logic systems can be enhanced compared with type-1 or type-2 fuzzy logic systems, the type reduction of type-n fuzzy sets (TnFS) will be more complex consequently. Mohammadzadeh,et al. have made deep researches on interval type-3 fuzzy logic systems (IT3FLS) and applied IT3FLS in fuzzy model identification [29]. Recently, IT3FLS has been widely applied in fuzzy control systems, like autonomous vehicles [30], frequency regulation system [31], multi-agent systems [32], micro-electro-mechanical-system gyroscopes [33], dynamic fractional-order models [34], MEMS gyroscopes [35], solar energy management systems [36], current sharing and voltage balancing in microgrids [37], predictive control [38 –40], parameterization of type-3 fuzzy controllers [41], nonlinear plants [42 –44],mobile robot [45 –47], speaker quality [48], image quality [49], telecom applications [50], chaotic systems [51], photovoltaic/battery systems [52], micro electro mechanical systems [53]. IT3FLS has also been applied in gas industry flowmeter fault detection [54], renewable energy modeling/prediction [55], COVID-19 time series prediction [56], modeling of CO₂ solubility [57], and Singh proposed an interval type-3 T-S fuzzy model in [58]. There are also some literatures on optimization of IT3FLS using Kalman filter, such as solve singular multi-pantograph equations [59] and self-organizing interval type-3 fuzzy logic systems [60].

Under the dual pressures of energy shortage and environmental pollution, more and more attentions have been paid to the development of efficient energy technology. Solid oxide fuel cells (SOFC) have been attracted extensive attentions due to their advantages of clean, pollution-free and high energy conversion efficiency. However, the characteristics of SOFC system are easily affected by operating and working conditions, such as ambient temperature, humidity, gas pressure and so on. It is not only quite complex, but also has strong nonlinearity. Some intelligent algorithms have been used for model identification of SOFC system, like artificial neural network model [61], hopfield neural network [62], extreme learning machines [63], BP neural networks [64].

For the nonlinear characteristics of SOFC system, the control strategy design is also an challenge. The main control variables of SOFC system are stack voltage and temperature, fuel utilization, hydrogen flow and power. The model predictive control was applied in integrated solid oxide fuel cell and turbocharger system, which increased the load transition speed and maintained the tack temperature within the allowable operation limits [65]. A nonlinear controller based on the unmolded dynamic compensation is developed to force the SOFC to track desired stack temperature and voltage [66]. Sun combined PI controller and a simple supervisory control strategy to provide a reasonable power reference for SOFC [67]. The control effect of output voltage when using fuzzy PID and conventional PID controller are compared in [68] and the flow rate of fuel (hydrogen) and air (oxygen) are optimized by output voltage control based on fuzzy controller [69]. In [70], four different control systems, PI controller, PI-PI cascade controller, PI-PI cascade controller with FF approaches, PID-PI cascade control with FF approaches are designed for cathode outlet temperature of a turbocharged solid oxide fuel cell system. Li, et al. proposed a SOFC output voltage data-driven controller based on multi-agent large-scale deep reinforcement learning [71]. Ghavidel, et al. proposed a robust observer-based control method for energy management strategy of hybrid fuel cell-battery-supercapacitor systems [72].

The main contributions of this paper are:

A direct type reduction for interval type-3 fuzzy sets (IT3FS) using NT type reduction algorithm is proposed, which can ensure the real-time of interval type-3 fuzzy controller.

An interval type-3 fuzzy PID control system is designed based on interval type-3 fuzzy logic systems.

The proposed interval type-3 fuzzy PID controller is applied in voltage control of SOFC systems.

2 Interval type-3 fuzzy sets

A general type-2 fuzzy sets $\tilde{A} (x)$ can be represented by the union of its associated type-2 fuzzy sets ${\tilde{A}}_{α} (x)$ as (1), with α ∈ [0, 1], ρ is number of α-planes. $\tilde{A} (x) = \frac{α_{1}}{{\tilde{A}}_{α_{1}} (x)} + \frac{α_{2}}{{\tilde{A}}_{α_{2}} (x)} + \dots + \frac{α_{ρ}}{{\tilde{A}}_{α_{ρ}} (x)}$ (1)

The secondary triangular membership function of a general type-2 fuzzy sets represented by α-plane can be described as Fig. 1. In Fig. 1, $[\underline{f} {\tilde{A}}_{α}, {\bar{f}}_{{\tilde{A}}_{α}}]$ are lower and upper membership degree of type-2 fuzzy sets ${\tilde{A}}_{α} (x)$ , whose secondary degree is α.

Fig. 1

General type-2 fuzzy sets represented by α-plane.

Compared with GT2FS, the secondary membership degrees of IT3FS are intervals, which is shown as (2).

$\begin{matrix} \tilde{A} (x) = {(x, u), [\underline{μ} \tilde{A} (x, u), {\bar{μ}}_{\tilde{A}} (x, u)] \\ | \forall x \in X, \forall u \in J_{x} \in [0, 1]} \end{matrix}$ (2)

By α-plane representation of IT3FS,the secondary membership degree of ${\tilde{A}}_{α} (x)$ α becomes an interval $\tilde{α} \in [\underline{α}, \bar{α}]$ , the horizontal slice representation of IT3FS $\tilde{A} (x)$ can be rewritten as (3).

$\begin{matrix} {\tilde{A}}_{\tilde{α}} (x) = {(x, u), \underline{α} ⩽ μ_{\tilde{A}} (x, u) ⩽ \bar{α} \\ | \forall x \in X, \forall u \in J_{x} \in [0, 1]} \end{matrix}$ (3)

If $\tilde{α} = ({\tilde{α}}_{1}, {\tilde{α}}_{2}, \dots, {\tilde{α}}_{ρ})$ , the IT3FS $\tilde{A} (x)$ can be defined as (4). $\tilde{A} (x) = \frac{1 / {\tilde{α}}_{1}}{{\tilde{A}}_{{\tilde{α}}_{1}} (x)} + \frac{1 / {\tilde{α}}_{2}}{{\tilde{A}}_{{\tilde{α}}_{2}} (x)} + \dots + \frac{1 / {\tilde{α}}_{ρ}}{{\tilde{A}}_{{\tilde{α}}_{ρ}} (x)}$ (4)

For $\tilde{α} \in [\underline{α}, \bar{α}]$ , (4) can be represented as (5). $\tilde{A} (x) = [\frac{1 / \underline{α} 1}{{\tilde{A}}_{\underline{α} 1} (x)}, \frac{1 / {\bar{α}}_{1}}{{\tilde{A}}_{{\bar{α}}_{1}} (x)}] + \dots + [\frac{1 / \underline{α} ρ}{{\tilde{A}}_{\underline{α} ρ} (x)}, \frac{1 / {\bar{α}}_{ρ}}{{\tilde{A}}_{{\bar{α}}_{ρ}} (x)}]$ (5)

3 Interval type-3 fuzzy PID control system

The structure scheme of a typical interval type-3 fuzzy control system (IT3FCS) is shown as Fig. 2. G_E and G_CE are scaling factors that transform the error and error derivative to interval type-3 fuzzy inference inputs E and $\dot{E}$ .

Fig. 2

Structure scheme of interval type-3 fuzzy control system.

In this paper, triangular primary membership function is adapted, and show as Figs. 3 4. E is defined in domain [L₁, L_m] and $\dot{E}$ is defined in domain [S₁, S_n].

Fig. 3

Primary membership function of E.

Fig. 4

Primary membership function of $\dot{E}$ .

From Figs. 3 4, the triangular primary membership function is orthogonal, consistent, complete and normal, that is:

${\bar{u}}_{{\tilde{A}}_{i}} (E) + {\bar{u}}_{{\tilde{A}}_{i + 1}} (E) = 1 and {\bar{u}}_{{\tilde{A}}_{k}} (E) = 0, i \in [1, m - 1], k \neq i, i + 1$

${\bar{u}}_{{\tilde{B}}_{j}} (\dot{E}) + {\bar{u}}_{{\tilde{B}}_{j + 1}} (\dot{E}) = 1 and {\bar{u}}_{{\tilde{B}}_{k}} (\dot{E}) = 0, j \in [1, n - 1], k \neq j, j + 1$

So, 4 fuzzy rules will be fired at one control period, these rules can be described as follows:

Rule 1: If E is ${\tilde{A}}_{i}$ and $\dot{E}$ is ${\tilde{B}}_{j}$ , then U = x₁

Rule 2: If E is ${\tilde{A}}_{i}$ and $\dot{E}$ is ${\tilde{B}}_{j + 1}$ , then U = x₂

Rule 3: If E is ${\tilde{A}}_{i + 1}$ and $\dot{E}$ is ${\tilde{B}}_{j}$ , then U = x₃

Rule 4: If E is ${\tilde{A}}_{i + 1}$ and $\dot{E}$ is ${\tilde{B}}_{j + 1}$ , then U = x₄

In an interval type-3 fuzzy logic systems under product operator, the lower and upper fired primary membership degree of fuzzy rule is a type-1 interval and can be defined as Equation (6). $[\underline{f_{1}}, {\bar{f}}_{1}] = [{\underline{u}}_{{\tilde{A}}_{i}} (E) \times {\underline{u}}_{{\tilde{B}}_{j}} (\dot{E}), {\bar{u}}_{{\tilde{A}}_{i}} (E) \times {\bar{u}}_{{\tilde{B}}_{j}} (\dot{E})]$ (6-1) $[\underline{f_{2}}, {\bar{f}}_{2}] = [{\underline{u}}_{{\tilde{A}}_{i}} (E) \times {\underline{u}}_{{\tilde{B}}_{j + 1}} (\dot{E}), {\bar{u}}_{{\tilde{A}}_{i}} (E) \times {\bar{u}}_{{\tilde{B}}_{j + 1}} (\dot{E})]$ (6-2) $[\underline{f_{3}}, {\bar{f}}_{3}] = [{\underline{u}}_{{\tilde{A}}_{i + 1}} (E) \times {\underline{u}}_{{\tilde{B}}_{j}} (\dot{E}), {\bar{u}}_{{\tilde{A}}_{i + 1}} (E) \times {\bar{u}}_{{\tilde{B}}_{j}} (\dot{E})]$ (6-3) $[\underline{f_{4}}, {\bar{f}}_{4}] = [{\underline{u}}_{{\tilde{A}}_{i + 1}} (E) \times {\underline{u}}_{{\tilde{B}}_{j + 1}} (\dot{E}), {\bar{u}}_{{\tilde{A}}_{i + 1}} (E) \times {\bar{u}}_{{\tilde{B}}_{j + 1}} (\dot{E})]$ (6-4)

In (6), $[\underline{f} k, {\bar{f}}_{k}]$ represent the lower and upper membership degree of ${\tilde{A}}_{α} (x)$ when α= 0. In this paper, triangular secondary membership function is adapted, which is shown as Fig. 5, the IT3FS will be a GT2FS when the fuzzification of IT3FS disappears.

Fig. 5

Secondary membership function of interval type-3 fuzzy sets.

Setting the number of α-plane is ρ and α _j (j = 1,2, . . . ,ρ) is average distributed in interval [0,1]. The apex of triangular secondary membership function can be defined as (7). $Ape (x_{k}) = \underline{f_{k}} + w \times ({\bar{f}}_{k} - \underline{f_{k}})$ (7) where, w is a constant value to determine the nature of secondary triangular membership function.

The type reduction of IT3FS can be implemented from GT2FS represented by α-plane. For IT3FS, the uncertainty of α _j is $[\underline{α_{j}}, {\bar{α}}_{j}]$ , where $\underline{α_{j}} = α_{j}^{Δ}$ and ${\bar{α}}_{j} = α_{j}^{1 / Δ}$ , the fuzziness parameter Δ ≥ 1 is a constant and represent the uncertainty of the vertical slice.

The lower and upper bounds of ${\bar{α}}_{j}$ -plane can be shown as (8). $\underline{f_{k}} ({\bar{α}}_{j}) = \underline{f_{k}} + {\bar{α}}_{j} \times (Ape (x_{k}) - \underline{f_{k}})$ (8-1) ${\bar{f}}_{k} ({\bar{α}}_{j}) = {\bar{f}}_{k} - {\bar{α}}_{j} \times ({\bar{f}}_{k} - Ape (x_{k}))$ (8-2)

The lower and upper bounds of $\underline{α} j$ -plane can be shown as (9). $\underline{f_{k}} (\underline{α} j) = {f_{k}}_{-}^{'} + \underline{α_{j}} \times (Ape (x_{k}) - {f_{k}}_{-}^{'})$ (9-1) ${\bar{f}}_{k} (\underline{α_{j}}) = {\bar{f}}_{k}^{'} - \underline{α_{j}} \times ({\bar{f}}_{k}^{'} - Ape (x_{k}))$ (9-2)

In (9), ${f_{k}}_{-}^{'}$ and ${\bar{f}}_{k}^{'}$ are the endpoints of footprint of uncertainty (FOU) for secondary triangular membership function. The FOU of secondary membership function can be excited by ɛ (0 ≤ ɛ < 0.5), that is, ${f_{k}}_{-}^{'} = \underline{f_{k}} + ɛ ({\bar{f}}_{k} - \underline{f_{k}})$ and ${\bar{f}}_{k}^{'} = {\bar{f}}_{k} - ɛ ({\bar{f}}_{k} - \underline{f_{k}})$ .

Because one α-plane is an interval type-2 fuzzy sets, so some type reduction algorithms for interval type-2 fuzzy sets, like KM,EKM, et al., can be applied for α-plane of GT2FS. For simplicity, a direct type reduction algorithm, NT type reduction using the average of lower and upper bounds for membership degree, will be used in this paper. Thus, the defuzzification result of ${\bar{α}}_{j}$ -plane and $\underline{α} j$ -plane can be calculated as (10) and (11). $U_{{\bar{α}}_{j}} = \frac{\sum_{k = 1}^{4} (\underline{f_{k}} ({\bar{α}}_{j}) + {\bar{f}}_{k} ({\bar{α}}_{j})) \times x_{k}}{\sum_{k = 1}^{4} (\underline{f_{k}} ({\bar{α}}_{j}) + {\bar{f}}_{k} ({\bar{α}}_{j}))}$ (10) $U_{\underline{α_{j}}} = \frac{\sum_{k = 1}^{4} (\underline{f_{k}} (\underline{α_{j}}) + {\bar{f}}_{k} (\underline{α_{j}})) \times x_{k}}{\sum_{k = 1}^{4} (\underline{f_{k}} (\underline{α_{j}}) + {\bar{f}}_{k} (\underline{α_{j}}))}$ (11)

The centroid of interval type-3 fuzzy inference can be represented as (12). $U = \frac{\sum_{j = 1}^{ρ} U_{{\bar{α}}_{j}} \times {\bar{α}}_{j} + \sum_{j = 1}^{ρ} U_{\underline{α_{j}}} \times \underline{α_{j}}}{\sum_{j = 1}^{ρ} {\bar{α}}_{j} + \sum_{j = 1}^{ρ} \underline{α_{j}}}$ (12)

The final controller output in Fig. 2 is calculated as (13). $u = G_{PD} \times U + G_{PI} \times \int U$ (13)

4 Simulations

In this section, the efficacy of proposed interval type-3 fuzzy PID control system is studied and applied in 2 nonlinear systems. Its performance and robustness will be compared with conventional PID control system (PIDCS), type-1 fuzzy control system (T1FCS),interval type-2 fuzzy control system (IT2FCS) and general type-2 fuzzy control system (GT2FCS). Here the same control system parameters are applied in each plant to display the robustness of T1FCS, IT2FCS, GT2FCS and IT3FCS.

For simplicity, primary membership functions of error and error derivative are identical, which are shown as Fig. 6. The consequent parameters are symmetry, where NB = - H, NM = - ηH, Z = 0, PM = ηH, PB = H in Table 1. In this paper, the coefficient η is fixed as 0.8, p1, p2 in Fig. 6 are fixed as 0.9 and 0.3, the number of α-plane ρ = 5.

Fig. 6

Triangular primary membership functions of E and $\dot{E}$ .

Table 1

Fuzzy rules of 4 fuzzy controllers

$E / \dot{E}$	NB	NM	Z	PM	PB
NB	NB	NB	NB	NM	Z
NM	NB	NB	NM	Z	PM
Z	NB	NM	Z	PM	PB
PM	NM	Z	PM	PB	PB
PB	Z	PM	PB	PB	PB

Table 1 lists the fuzzy rules of T1FCS, IT2FCS, GT2FCS and IT3FCS.

4.1 Second order nonlinear plant (P1)

$\frac{d^{2} y (t)}{{dt}^{2}} + 2 \frac{dy (t)}{dt} + y^{2} (t) = u (t)$

PID controlled parameters are K_P = 0.8028, K_I = 1.8548, K_D = 0.4609. The parameters of 4 fuzzy PID controllers are G_E = 0.5, G_CE = 0.4, G_PD = 0.2937, G_PI = 3.228. The membership function parameters of plant 1 are listed in Table 2.

Table 2
Membership function parameters of P1

Ep H w Δ ɛ

1 1 10 8 0.3

Ep	H	w	Δ	ɛ
1	1	10	8	0.3

Figure 7 show the system output curves of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS for plant 1.

Fig. 7

The system output curves of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS for plant 1.

Table 3 summarizes some control performance comparisons of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS. And control performance indexes include steady state time (t_s, that is time when the absolute value of system error reached the 2% of set value), rising time (t_r, that is time when the system output reach the 2% of set value), overshoot (OS), three error integral criterions (that is an integral function of error between the set value and system output under the unit step disturbance), mainly contain ISE, ITSE, IAE,ITAE, which is defined as follows. In Table 3, the simulation time is 20 s.

Table 3

Control performance comparisons of IT3 fuzzy controller with other controllers (P1)

Performance index	PID	T1	IT2	GT2	IT3
t_s (s)	3.6	3.8	3.8	3.8	3.8
t_r (s)	12.8	7.6	8.4	6	5.9
OS (%)	34.1	16.73	19.51	10.08	9.56
ISE	2.3894	2.2907	2.3502	2.2514	2.2497
ITSE	4.0961	3.0289	3.2925	2.8158	2.8074
IAE	3.6935	2.9973	3.1875	2.7487	2.7346
ITAE	11.3166	5.8695	7.0040	4.4303	4.3562

$ISE = \int_{0}^{ts} e (t)^{2} dt$ $ITSE = \int_{0}^{ts} t \times e (t)^{2} dt$ $IAE = \int_{0}^{ts} | e (t) | dt$ $ITAE = \int_{0}^{ts} t \times | e (t) | dt$

4.2 Nonlinear inverted pendulum plant (P2)

The state equations of the inverted pendulum system can be expressed as follow. $\begin{matrix} [\begin{matrix} {\dot{x}}_{1} \\ {\dot{x}}_{2} \end{matrix}] = [\begin{matrix} x_{2} \\ \frac{g sin (x_{1}) - \frac{m_{p} {lx}_{2}^{2} sin (x_{1}) cos (x_{1})}{(m_{p} + m_{c})}}{\frac{4 l}{3} - \frac{m_{p} l cos (x_{1})^{2}}{(m_{p} + m_{c})}} \end{matrix}] \\ + [\begin{matrix} 0 \\ \frac{\frac{cos (x_{1})}{(m_{p} + m_{c})}}{\frac{4 l}{3} - \frac{m_{p} l cos (x_{1})^{2}}{(m_{p} + m_{c})}} \end{matrix}] u \end{matrix}$ where, x₁ = θ is the angle of the pendulum and $x_{2} = {\dot{x}}_{1} = \dot{θ}$ is angular velocity of the pendulum.

u is the control force in the unit (Newton) applied horizontally to the cart. m_c is the mass of the cart, m_p is the mass of the pendulum and l is the half length of the pendulum. The values of these parameters are m_c = 0.5 kg, m_c = 0.2 kg, l = 0.5 m, g = 9.8 m/s².

PID controlled parameters are K_P = 40, K_I = 100, K_D = 8. The parameters of 4 fuzzy PID controllers are G_E = 0.1009, G_CE = 0.1944, G_PD = 30.5501, G_PI = 30.2681. The membership function parameters of plant 2 are listed in Table 4.

Table 4
Membership function parameters of P2

Ep H w Δ ɛ

0.2 4 0.08 1.94 0.2

Ep	H	w	Δ	ɛ
0.2	4	0.08	1.94	0.2

Figure 8 show the system output curves of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS for plant 2.

Fig. 8

The system output curves of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS for plant 2.

Table 5 shows the P2 control performance comparisons of PIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS. In Table 5, the simulation time is 3 s.

Table 5

Control performance comparisons of IT3 fuzzy controller with other controllers (P2)

Performance index	PID	T1	IT2	GT2	IT3
ISE (×10^–4)	2.7853	2.3148	1.6565	1.5648	1.5657
ITSE (×10^–6)	23.449	13.463	3.9005	2.3108	2.2953
IAE	0.0101	0.0074	0.0040	0.0026	0.0025
ITAE (×10^–4)	36	14	5.6539	2.5692	2.4618
RMSE	0.0096	0.0088	0.0074	0.0072	0.0072

5 Fuel cell power plant

The charge state adjustment of solid oxide fuel cell has an important impact for the safe operation of battery system and the quality of battery power supply. However, charge state adjustment system is quite complex and the controlled object also has strong nonlinearity. Qin, et al. in [68] has described nonlinear model of a fuel cell power plant (FCPP) in detail. In this paper, the proposed interval type-3 fuzzy PID controller will be used in the fuel cell power plant described in [68].

PID controlled parameters are K_P = 0.3314, K_I = 0.0201, K_D = 0.4632. The parameters of 4 fuzzy PID controllers are G_E = 0.12, G_CE = 0.12, G_PD = 29.1314, G_PI = 0.29801. The membership function parameters of FCPP are listed in Table 6.

Table 6
Membership function parameters of FCPP

Ep H w Δ ɛ

5 0.5 10 2 0.1

Ep	H	w	Δ	ɛ
5	0.5	10	2	0.1

5.1 Voltage step response

a) Normal case

For comparing with adaptive fuzzy PID control system (AFPIDCS)in [68], the current is kept at 300 A, the rated voltage begins with 305 V, and then it is set to increase by 15 V at 100 s, 200 s, 300 s, and 400 s. Then the rated voltage decreases by 30 V at 500 s and 600 s to return to 305 V, the initial voltage is 330 V. In Qin’s conclusions, the control effect of AFPIDCS is better than PID control system, only the IT3FCS and AFPIDCS will be compared. Figure 9 show the voltage output curves of IT3FCS and AFPIDCS for step response of fuel cell power plant and Fig. 10 shows the fuel flow curves.

Fig. 9

The voltage output curves of IT3FCS and AFPIDCS under voltage step response.

Fig. 10

The fuel flow curves of IT3FCS and AFPIDCS under voltage step response.

Table 7 shows the fuel cell power plant control performance comparisons of PIDCS, AFPIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS under voltage step response, the simulation time is 700 s.

Table 7

Control performance comparisons of IT3 fuzzy controller with other controllers for FCPP under voltage step response

Performance index	PID	AFPID	T1	IT2	GT2	IT3
ISE	13864.9532	12975.0889	12162.8528	12204.9544	12050.38	12048.8896
ITSE	78111.9311	51246.94059	39505.3299	36710.1264	35625.57	34511.0427
IAE	1303.9998	981.9677	898.9283	959.209	798.7837	797.4562
ITAE	20413.9818	11713.6852	12210.0817	15020.0235	7584.983	7523.7247

b) Fuel flow disturbance response

In this case, the fuel flow adding disturbance will be discussed. Figure 11 show the voltage output curves of IT3FCS and AFPIDCS under voltage step response of fuel cell power plant with fuel flow disturbance and Fig. 12 shows the fuel flow curves.

Fig. 11

The voltage output curves of IT3FCS and AFPIDCS under voltage step response with fuel flow disturbance.

Fig. 12

The fuel flow curves of IT3FCS and AFPIDCS under voltage step response with fuel flow disturbance.

Table 8 shows the fuel cell power plant control performance comparisons of PIDCS, AFPIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS under voltage step response with fuel flow disturbance.

Table 8

Control performance comparisons of IT3 fuzzy controller with other controllers for FCPP under voltage step response with fuel flow disturbance

Performance index	PID	AFPID	T1	IT2	GT2	IT3
ISE	13864.89	12975.1	12162.86	12204.96	12050.39	12048.9
ITSE	78106.89	51244.52	39503.58	42668.2	35625.2	35591.58
IAE	1303.998	981.8898	898.882	959.1404	798.7725	797.4452
ITAE	20414.35	11709.31	12208.09	15017.38	7584.524	7523.272

5.2 Current disturbance response

a) Normal case

In this section, the rated voltage is kept at 305 V, and the current starts with 300A, then the current is set to increase by 75A at 50 s, 100 s, 150 s, and 200 s and then decreases by 150A at 250 s and 300 s to return to 300A. Figure 13 show the voltage output curves of IT3FCS and FPIDCS under current disturbance response of fuel cell power plant and Fig. 14 shows the fuel flow curves.

Fig. 13

The voltage output curves of IT3FCS and AFPIDCS under current disturbance response.

Fig. 14

The fuel flow curves of IT3FCS and AFPIDCS under current disturbance response.

Table 9 shows the fuel cell power plant control performance comparisons of PIDCS, AFPIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS under current disturbance response, the simulation time is 350 s.

Table 9

Control performance comparisons of IT3 fuzzy controller with other controllers for FCPP under current disturbance response

Performance index	PID	AFPID	T1	IT2	GT2	IT3
ISE	4482.3454	4839.5515	4292.6284	4448.9261	4107.8286	4106.06
ITSE	935550.5923	878836.7362	878638.6968	892224.4496	864801.0301	864654.9
IAE	648.8442	595.5169	538.8995	611.584	438.3204	437.2982
ITAE	122946.0826	88066.8186	88783.9632	98673.29	78420.7407	78343.33

b) Fuel flow disturbance response

In this case, the fuel flow adding disturbance will be discussed. Figure 15 show the voltage output curves of IT3FCS and FPIDCS under current disturbance response of fuel cell power plant with fuel flow disturbance and Fig. 16 shows the fuel flow curves.

Fig. 15

The voltage output curves of IT3FCS and AFPIDCS under current disturbance response with fuel flow disturbance.

Fig. 16

The fuel flow curves of IT3FCS and AFPIDCS under current disturbance response with fuel flow disturbance.

Table 10 shows the fuel cell power plant control performance comparisons of PIDCS, AFPIDCS, T1FCS, IT2FCS, GT2FCS and IT3FCS under current disturbance response with fuel flow disturbance.

Table 10

Control performance comparisons of IT3 fuzzy controller with other controllers for FCPP under current disturbance response with fuel flow disturbance

Performance index	PID	AFPID	T1	IT2	GT2	IT3
ISE	4482.2244	4839.496	4292.6688	4448.989	4107.8476	4106.0787
ITSE	935515.381	878837.9798	878651.8901	892247.592	864805.0674	864658.8248
IAE	648.8141	595.4979	538.9417	611.6374	438.326	437.3038
ITAE	122939.785	88064.5699	88798.6941	98692.1442	78424.2311	78346.7584

6 Conclusions

As interval type-3 fuzzy sets (IT3FS) are derived from general type-2 fuzzy sets, the secondary membership degree of IT3FS is an interval and its type reduction procedure can be implemented as type reduction of GT2FS represented by α-plane. Unlike general type-2 fuzzy sets that α is crisp value, the α of interval type-3 fuzzy sets becomes an interval, which makes the interval type-3 fuzzy has more design freedoms than type-2 fuzzy sets.

In this paper, an interval type-3 fuzzy PID (IT3FPID) controller is proposed firstly. The control effects of IT3FPID controller are compared with tradition PID controller, type-1 fuzzy (T1FPID) controller, interval type-2 fuzzy controller (IT2FPID) and general type-2 fuzzy PID (GT2FPID) controller. The simulation results of 2 nonlinear plants show that the proposed IT3FPID controller can reduce system overshoot, improve response speed and reduce steady state time. Then the IT3FPID controller is applied in a solid oxide fuel cells power plant to control the rated output voltage. Two conditions are tested, the one is desired rated voltage changed and the other is the current contains disturbance. The control preferences are also compared with AFPID controller in [68], which indicate that IT3FPID can achieve better control effects than AFPID controller.

The future researched will focus on the following 2 aspects:

The IT3FPID controller contains more parameters, some parameters are fixed in this paper and how to chose the best parameters is a changing issue.

Apply the proposed IT3FCS in practical solid oxide fuel cells power plant.

References

Mendel

J.M.

, Uncertain rule based fuzzy logic systems: introduction and new directions (2nd edition), Springer, 2017.

Karnik

N.N.

and Mendel

J.M.

, Centroid of a type-2 fuzzy set, Information Sciences 132 (2001), 195–220.

Mendel

J.M.

, General type-2 fuzzy logic systems made simple: a tutorial, IEEE Transactions on Fuzzy Systems 22 (2014), 1162–1182.

Torshizi

A.D.

, Zarandi

M.H.F.

and Zakeri

, On type-reduction of type-2 fuzzy sets: A review, Applied Soft Computing 27 (2015), 614–627.

Golsefid

S.M.M.

, Zarandi

M.H.F.

and Turksen

I.B.

, Multi-central general type-2 fuzzy clustering approach for pattern recognitions, Information Sciences 328 (2016), 172–188.

Hassani

, Zarei

, Arefi

M.M.

and Razavi-Fa

, zSlices-based general type-2 fuzzy fusion of support vector machines with application to bearing fault detection, IEEE Transactions on Industrial Electronics 64 (2017), 7210–7217.

Andreu-Perez

, et al., A self-adaptive online brain-machine interface of a humanoid robot through a general type-2 fuzzy inference system, IEEE Transactions on Fuzzy Systems 26 (2018), 101–116.

Rubio-Solis

, et al., General type-2 radial basis function neural network: a data-driven fuzzy model, IEEE Transactions on Fuzzy Systems 27 (2019), 333–347.

Sakalli

, Kumbasar

and Mendel

J.M.

, Towards systematic design of general type-2 fuzzy logic controllers: analysis, interpretation, and tuning, IEEE Transactions on Fuzzy Systems 29 (2021), 226–239.

10.

, Mahfouf

and Torres-Salomao

L.A.

, An adaptive general type-2 fuzzy logic approach for psychophysiological state modeling in real-time human-machine interfaces, IEEE Transactions on Human-Machine Systems 51 (2021), 1–11.

11.

Mansoureh

, Hossein

F.Z.M.

and Susan

, A multilayer general type-2 fuzzy community detection model in large-scale social networks, IEEE Transactions on Fuzzy Systems 30 (2022), 4494–4503.

12.

Laha

, et al., Hemodynamic analysis for olfactory perceptual degradation assessment using generalized type-2 fuzzy regression, IEEE Transactions on Cognitive and Developmental Systems 14 (2022), 1217–1231.

13.

Kumbasar

and Hagras

, A self-tuning zSlices-based general type-2 fuzzy PI controller, IEEE Transactions on Fuzzy Systems 23 (2015), 991–1013.

14.

Sanchez

M.A.

, Castillo

and Castro

J.R.

, Generalized type-2 fuzzy systems for controlling a mobile robot and a performance comparison with Interval type-2 and type-1 fuzzy systems, Expert Systems with Applications 42 (2015), 5904–5914.

15.

Castillo

, et al., 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 (2016), 257–274.

16.

Khooban

M.H.

, et al., An optimal general type-2 fuzzy controller for urban traffic network, ISA Transactions 66 (2016), 335–343.

17.

Ontiveros

, Melin

and Castillo

, High order α-planes integration: a new approach to computational cost reduction of general type-2 fuzzy systems, Engineering Applications of Artificial Intelligence 74 (2018), 186–197.

18.

Baghbani

, Akbarzadeh-T

M.-R.

and Akbarzadeh

, Indirect adaptive robust mixed H²/H^∞ general type-2 fuzzy control of uncertain nonlinear systems, Applied Soft Computing 72 (2018), 392–418.

19.

Mohammadzadeh

and Kaynak

, A novel general type-2 fuzzy controller for fractional-order multi-agent systems under unknown time-varying topology, Journal of the Franklin Institute 36 (2019), 5151–5171.

20.

Zhao

, et al., Non-singleton general type-2 fuzzy control for a two-wheeled self-balancing robot, International Journal of Fuzzy Systems 21 (2019), 1724–1737.

21.

Dian

S.Y.

, et al., Double closed-loop general type-2 fuzzy sliding model control for trajectory tracking of wheeled mobile robots, International Journal of Fuzzy Systems 21 (2019), 2032–2042.

22.

Khooban

M.H.

, Niknam

and Sha-Sadeghi

, A time-varying general type-II fuzzy sliding mode controller for a class of nonlinear power systems, Journal of Intelligent & Fuzzy Systems 30 (2016), 2927–2937.

23.

Castillo

, et al., A generalized type-2 fuzzy granular approach with applications to aerospace, Information Sciences 354 (2016), 165–177.

24.

Cervantes

and Castillo

, Type-2 fuzzy logic aggregation of multiple fuzzy controllers for airplane flight control, Information Sciences 324 (2015), 247–256.

25.

Shi

J.Z.

, et al., An improved general type 2 fuzzy sets type reduction and its application in general type 2 fuzzy controller design, Soft Computing 23 (2019), 13513–13530.

26.

Shi

J.Z.

, 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 (2022), 1808–1824.

27.

Zhao

, et al., General type-2 fuzzy gain scheduling PID controller with application to power-line inspection robots, International Journal of Fuzzy Systems 22 (2020), 181–200.

28.

Rickard

J.T.

, Aisbett

and Gibbon

, Fuzzy subsethood for fuzzy sets of type-2 and generalized type-n, IEEE Transactions on Fuzzy Systems 17 (2009), 50–60.

29.

Mohammadzadeh

, Sabzalian

M.H.

and Zhang

, An interval type-3 fuzzy system and a new online fractional-order learning algorithm: theory and practice, IEEE Transactions on Fuzzy Systems 28 (2020), 1940–1950.

30.

Tian

W.W.

, et al., Stability of interval type-3 fuzzy controllers for autonomous vehicles, Mathematics 9 (2021), 2742.

31.

Aly

A.A.

, et al., Frequency regulation system: a deep learning identification,type-3 fuzzy control and LMI stability analysis, Energies 14 (2021), 7801.

32.

Yan

, et al., A new event-triggered type-3 fuzzy control system for multi-agent systems: optimal economic efficient approach for actuator activating, Electronics 10 (2021), 3122.

33.

Alattas

K.A.

, et al., A new data-driven control system for MEMSs gyroscopes: dynamics estimation by type-3 fuzzy systems, Micromachines 12 (2021), 1390.

34.

Mohammadzadeh

, et al., A novel fractional-order multiple-model type-3 fuzzy control for nonlinear systems with unmodeled dynamics, International Journal of Fuzzy Systems 23 (2021), 1633–1651.

35.

Vafaie

R.H.

, Mohammadzadeh

and Piran

M.J.

, A new type-3 fuzzy predictive controller for MEMS gyroscopes, Nonlinear Dynamics 106 (2021), 381–403.

36.

Liu

, et al., A new online learned interval type-3 fuzzy control system for solar energy management systems, IEEE Access 9 (2021), 10498–10508.

37.

Taghieh

, et al., A type-3 fuzzy control for current sharing and voltage balancing in microgrids, Applied Soft Computing 129 (2022), 109636.

38.

Gheisarnejad

, Mohammadzadeh

and Khooban

M.H.

, Model predictive control based type-3 fuzzy estimator for voltage stabilization of DC power converters, IEEE Transactions on Industrial Electronics 69 (2022), 13849–13858.

39.

Taghieh

, et al., A predictive type-3 fuzzy control for underactuated surface vehicles, Ocean Engineering 266 (2022), 113014.

40.

Taghieh

, et al., A hybrid predictive type-3 fuzzy control for time-delay multi-agent systems, Electronics 11 (2022), 63.

41.

Ochoa1

, et al., Interval type-3 fuzzy differential evolution for parameterization of fuzzy controllers, International Journal of Fuzzy Systems 25 (2023), 1360–1376.

42.

Amador-Angulo

, et al., A new approach for interval type-3 fuzzy control of nonlinear plants, International Journal of Fuzzy Systems 25 (2023), 1624–1642.

43.

Yan

B.C.

, Generation of limit cycles in nonlinear systems: machine leaning based type-3 fuzzy control, IEEE Access 11 (2023), 34835–34845.

44.

Taghieh

, et al., A novel adaptive interval type-3 neuro-fuzzy robust controller for nonlinear complex dynamical systems with inherent uncertainties, Nonlinear Dynamics 111 (2023), 411–425.

45.

Alkabaa

A.S.

, A practical type-3 fuzzy control for mobile robots: predictive and Boltzmann-based learning, Complex & Intelligent Systems (2023). https://doi.org/10.1007/s40747-023-01086-4

46.

Peraza

, et al., Interval-type 3 fuzzy differential evolution for designing an interval-type 3 fuzzy controller of a unicycle mobile robot, Mathematics 10 (2022), 3533.

47.

Amador-Angulo

, et al., Interval type-3 fuzzy adaptation of the bee colony optimization algorithm for optimal fuzzy control of an autonomous mobile robot, Micromachines 13 (2022), 1490.

48.

Castillo

, Castro

J.R.

and Melin

, Interval type-3 fuzzy fractal approach in sound speaker quality control evaluation, Engineering Applications of Artificial Intelligence 116 (2022), 105363.

49.

Castillo

, Castro

J.R.

and Melin

, Interval type-3 fuzzy control for automated tuning of image quality in televisions, Axioms 11 (2022), 276.

50.

Gheisarnejad

, et al., Stabilization of 5G telecom converter-based deep type-3 fuzzy machine learning control for telecom applications, IEEE Transactions on Circuits and Systems II: Express Briefs 69 (2022), 544–548.

51.

Balootaki

M.A.

, et al., Non-singleton fuzzy control for multi-synchronization of chaotic systems, Applied Soft Computing 99 (2021), 106924.

52.

Mosavi

, et al., Fractional-order fuzzy control approach for photovoltaic/battery systems under unknown dynamics, variable irradiation and temperature, Electronics 9 (2020), 1455.

53.

Mohammadzadeh

and Vafaie

R.H.

, A deep learned fuzzy control for inertial sensing: Micro electro mechanical systems, Applied Soft Computing 109 (2021), 107597.

54.

Wang

J.H.

, et al., Non-singleton type-3 fuzzy approach for flowmeter fault detection: experimental study in a gas industry, Sensors 21 (2021), 7419.

55.

Cao

, et al., Mosavi. Deep learned recurrent type-3 fuzzy system: application for renewable energy modeling/prediction, Energy Reports 7 (2021), 8115–8127.

56.

Castillo

, et al., Interval type-3 fuzzy aggregators for ensembles of neural networks in COVID-19 time series prediction, Engineering Applications of Artificial Intelligence 114 (2022), 105110.

57.

Tian

M.W.

, et al., A deep-learned type-3 fuzzy system and its application in modeling problems, Acta Polytechnica Hungarica 19 (2022), 151–172.

58.

Singh

D.J.

, et al., An approach towards the design of interval type-3 T-S fuzzy system, IEEE Transactions on Fuzzy Systems 30 (2022), 3880–3893.

59.

, et al., Optimal type-3 fuzzy system for solving singular multi-pantograph equations, IEEE Access 8 (2020), 225692–225702.

60.

Qasem

S.N.

, et al., A type-3 logic fuzzy system: optimized by a correntropy based Kalman filter with adaptive fuzzy kernel size, Information Sciences 572 (2021), 424–443.

61.

Razbani

and Assadi

, Artificial neural network model of a short stack solid oxide fuel cell based on experimental data, Journal of Power Sources 246 (2014), 581–586.

62.

S.S.

, Xia

and Gibbons

E.M.

, Model identification and strategy application for solid oxide fuel cell using rotor hopfield neural network based on a novel optimization method, International Journal of Hydrogen Energy 45 (2020), 27694–27704.

63.

Zhang

, et al., An optimal model identification for solid oxide fuel cell based on extreme learning machines optimized by improved red fox optimization algorithm, International Journal of Hydrogen Energy 46 (2021), 28270–28281.

64.

L.M.

, et al., A novel multi-physics and multi-dimensional model for solid oxide fuel cell stacks based on alternative mapping of BP neural networks, Journal of Power Sources 500 (2021), 229784.

65.

S.-R.

, et al., Model predictive control for power and thermal management of an integrated solid oxide fuel cell and turbocharger System, IEEE Transactions on Control Systems Technology 22 (2014), 911–920.

66.

X.J.

, et al., Control of a solid oxide fuel cell stack based on unmodeled dynamic compensations, International Journal of Hydrogen Energy 43 (2018), 22500–22510.

67.

Sun

, et al., Coordinated control strategies for fuel cell power plant in a microgrid, IEEE Transactions on Energy Conversion 33 (2018), 1–9.

68.

Qin

Y.X.

, et al., A fuzzy adaptive PID controller design for fuel cell power plant, Sustainability 10 (2018), 2438.

69.

Darjat , et al., Designing hydrogen and oxygen flow rate control on a solid oxide fuel cell simulator using the fuzzy logic control method, Processes 8 (2020), 154.

70.

Mantelli

, Ferrari

M.L.

and Traverso

, Dynamics and control of a turbocharged solid oxide fuel cell system, Applied Thermal Engineering 191 (2021), 116862.

71.

J.W.

, Yu

and Yang

, A data-driven output voltage control of solid oxide fuel cell using multi-agent deep reinforcement learning, Applied Energy 304 (2021), 117541.

72.

Ghavidel

H.F.

and Mousavi-G

S.M.

, Modeling analysis, control, and type-2 fuzzy energy management strategy of hybrid fuel cell-battery-supercapacitor systems, Journal of Energy Storage 51 (2022), 104456.

An interval type-3 fuzzy PID control system design and its application in solid oxide fuel cells power plant

Abstract

Keywords

1 Introduction

2 Interval type-3 fuzzy sets

Table 2 Membership function parameters of P1 Ep H w Δ ɛ 1 1 10 8 0.3

Table 4 Membership function parameters of P2 Ep H w Δ ɛ 0.2 4 0.08 1.94 0.2

Table 6 Membership function parameters of FCPP Ep H w Δ ɛ 5 0.5 10 2 0.1

References

Table 2
Membership function parameters of P1

Ep H w Δ ɛ

1 1 10 8 0.3

Table 4
Membership function parameters of P2

Ep H w Δ ɛ

0.2 4 0.08 1.94 0.2

Table 6
Membership function parameters of FCPP

Ep H w Δ ɛ

5 0.5 10 2 0.1