Actuator and system component fault tolerant control using interval type-2 Takagi-Sugeno fuzzy controller for hybrid nonlinear process

Abstract

The paper present efficient Fault-tolerant Control approach using Interval Type-2 Takagi-Sugeno Fuzzy Controller (IT2TSFLC) with conventional Proportional Integral (PI) controller for MIMO hybrid nonlinear level process. The implementation of control algorithms for MIMO systems is challenging due to variations in dynamics because of changing in operating point and the characteristics of nonlinear dynamic interaction. Such difficulties often deteriorate the performance of industrial MIMO process. This work develops T-S modeling of MIMO hybrid level process thereafter designed FTC using combination of knowledge base IT2TSFLC controller and model base PI controller subject to actuator and system component faults. To verify the applicability of the proposed FTC, this work presents a Two Conical Tank Interacting Level System (TCTILS) to examine the control performance. Along with, simulation results of proposed scheme is compared with Type-1 Fuzzy Logic Controller (T1FLC) and PID controller. Simulation results show that the IT2TSFLC relies a good dynamic behaviour of the TCTILS, a perfect level tracking with no overshoot, lesser settling and rise time. Two different error indices Integral Absolute Error (IAE) and Integral Square Error (ISE) are used to validate the proposed FTC scheme among two other scheme.

Keywords

Actuator fault interval type-2 T-S fuzzy logic control nonlinear system MIMO system PID controller system component fault Two Conical Tank Interacting Level System

1. Introduction

In any industries accuracy, profitability, availability and human safety are critical parameters. To maintain these parameters efficient and accurate control algorithms are needed, since past three decades different control algorithms ware developed for various engineering applications presented in [1, 2, 3, 4, 5]. Fault-tolerant Control (FTC) is an advance control scheme, which repel the effects of the faults on the control performance and maintain control performance at acceptable level with overall system stability [6]. Hence from past two decades, the importance of FTC systems becomes increasingly apparent, and considerable amount of research has been done in this area [5, 7, 8, 9, 10, 11, 12, 13, 14]. To tolerate the fault effects on system performance two redundancy ware used, one is hardware and second on is analytical redundancy. The hardware redundancy is not practical solution for any engineering system due to high cost and complexity of the system [15]. To overcome this limitation an analytical (computational) redundancy is right selection. The advantage of the analytical redundancy method is that it makes possible to identify the occurrence of the fault and to deal with the unsafe situation using the information of the extent and class of the faults. To design point of view FTC scheme has classified in two types: Active Fault Tolerant Control (AFTC) and Passive Fault Tolerant Control (PFTC).

The AFTC method is based on the fact that the controller can reset its parameters or even change its structure, as well as implement rapid dynamic compensation control output to keep system stable after the fault occurs. To overcome this problem, the AFTC method must have the ability to actively acquire the information of the faults or system state changes. Currently, Fault Diagnosis and Isolation (FDI) approach is an important strategy for the design of AFTC. The AFTC is overly dependent on the FDI performance, and it may lead to failure of fault-tolerant control system, due to the mistake detection, omission, long delay or large diagnosis error caused by fault diagnosis mechanism [6, 13]. The PFTC method is based on robust control ideas. It attributes the fault to the model parameter perturbation issue, and compensates the fault by the kind of fixed gain controller with strong robustness to ensure the system is insensitive to certain faults [12]. It can be found that the PFTC method require designers to predict as many faults as possible before the controller design. According to the changing characteristics of the dynamic system model parameters in the fault states, a robust controller with fixed structure and gain are worked out, which it does not need to be adjusted on-line for the particular predicted fault modes. Its advantage is that the control law is fixed and easy to implement, and it can guarantee the system stability and achieve the preset performance whether or not a fault. However, the controller must calculate the parameters for the most serious fault condition that the system can take. It often lead to the control law too conservative, and with the increase of the system fault complexity, the controller design process will become more difficult. According to a variety and severity of different faults (i.e. abrupt and incipient fault nature) that may affect the system, different levels of performance have to be considered in different fault scenarios. In addition, to guarantee that the closed-loop system be able to track a reference trajectory even in the event of faults. In the case of performance degradation, actuator faults (loss of effectiveness) and sensor faults avoidance being required, design controller needs to adjust the output that maintain the acceptable control performance and system stability in the event of faults [21].

In recent years, because of its combination of continuous and discrete dynamics researchers are more and more attracted to develop an efficient control strategies for SISO multi-tank as well as MIMO multi-tank level system [1, 2, 12, 13]. To the best of our knowledge, few works have published on the topic of fuzzy logic based fault tolerant system for interacting and non-interacting level control process. However, there are still many open issues which have to be addressed.

Although traditional fuzzy theory can resolve a number of problems, its robustness can be improved. Some researchers have proposed interval type-2 T-S fuzzy theory to improve the robustness of the fuzzy control systems. Type-2 fuzzy sets and their related definitions have been provided in related literature [16, 17, 18, 19]. In [1, 2, 20] authors are investigate the Passive FTC responses for interacting and non-interacting level control system for actuator, sensor and system component faults using fuzzy logic control. In [21, 22, 23] authors proposed and simulate the FTC for highly nonlinear level control system (i.e. two-tank non-interacting conical tank system) subject to actuator and system component faults, and found significant results. But less research and development on FTC using Interval Type-2 fuzzy Control system has been done before. Due to the advantages of Interval Type-2 fuzzy system and TS fuzzy dynamic model, a FTC based on interval Type-2 TS fuzzy control is proposed in this paper. It can show a great potential in handling various modeling as well as control applications.

This paper proposes a FTC strategy to tolerate the actuator and system component faults for interacting level control system based on Interval Type-2 T-S Fuzzy logic control (IT2FLC). In the present work, the capabilities of PFTC to handle two faults and reference trajectory management are exploited to design a fault tolerant controller. The main contribution of this paper is to introduce a novel FTC strategy based on IT2FLC for interacting level control process subject to actuator and system component faults. The simulation results are shown. The paper is organized as follows. Design of FTC design using IT2FLC and PID controller are considered in Section 2. Problem description and mathematical model is detailed in Section 3. Section 4 is dedicated to the simulation results with different types of fault and magnitude, followed by conclusions in Section 5.

Figure 1.

Type-2 FLS structure block diagram [22].

2. FTC design using interval type-2 T-S fuzzy logic controller

2.1 Background of the type-2 fuzzy logic control

The concept of type-II fuzzy sets was also proposed by [24] to overcome this limitation [24]. However, [25] has provided more insight by developing and presenting the first type-II fuzzy logic system (T2FLS) in [25]. T2FLCs are able to outperform their colleague T1FLC, since their fuzzy sets are characterized by membership functions (MFs) which themselves are fuzzy compared to fuzzy sets of T1FLS, containing MFs having crisp values. Type-II fuzzy sets consists of Foot-print of Uncertainty (FoU), it is a bounded region of the uncertainty in the primary membership grades of a type-II MF. It is the union of all primary membership grades. Which makes it generally more robust and well capable to remove high oscillations. The interval type-II fuzzy logic system (IT2FLS), currently the most extensively used for their minimal computational cost is a special case of type-II FLS [26]. Type-1 and type-2 fuzzy logic are mainly similar. The only essential difference between them which is the membership functions shape, besides the output process. Indeed, an interval type-2 fuzzy controller is consisting of: a fuzzifier, an inference engine, a rules base, a type reduction and a defuzzifier [4, 27, 28, 29, 30] the block diagram of the type-2 FLC is presented in Fig. 1.

2.2 Type-2 fuzzy logic control

2.2.1 Fuzzifier

The fuzzifier maps the crisp input vector $(e_{1}$ , $e_{2}$ …, $e_{n})^{T}$ to a type-2 fuzzy system $\tilde{A}_{x}$ , which very similar to the procedure performed in a type-1 fuzzy logic system.

Rules

The general form of the $i^{\text{th}}$ rule of the type-2 fuzzy logic system can be written as:

$\displaystyle\text{If }e_{1}\text{ is }\tilde{F}_{1}^{i}\text{ and }e_{2}\text% { is }\tilde{F}_{2}^{i}\text{ and }e_{n}\text{ is }\tilde{F}_{n}^{i},$

(1) $\displaystyle\text{then }y^{i}=\tilde{G}^{i}i=1,\ldots,M$

where: $\tilde{F}_{j}^{i}$ is represents the type-2 fuzzy system of the input state j of the $i^{\text{th}}$ rule, $x_{1}$ , $x_{2}$ , …, $x_{n}$ are the inputs, $\tilde{G}^{i}$ is the output of the type-2 fuzzy system for the rule $i$ , and $M$ is the number of rules. As can be seen, the rule structure of type-2 fuzzy logic system is similar to type-1 fuzzy logic system except that type-1 membership functions are replaced by their type-2 counterparts.
2.2.2 Inference engine

In a fuzzy interval type-2 using the minimum or product $t$ -norms operations, the $i^{\text{th}}$ activated rule $F^{i}(x_{1}$ , … $x_{n})$ produces the interval that is determined by two extremes $\underline{f}^{i}(x_{1},\ldots x_{n})$ and $\overline{f}^{i}(x_{1},\ldots x_{n})$ like written below [26]:

$\displaystyle\!\!\!\!\!\!\!\!F^{i}(x_{1},\ldots x_{n})=\left[\underline{f}^{i}% (x_{1},\ldots x_{n}),\overline{f}^{i}(x_{1},\ldots x_{n})\right]\doteq\left[% \underline{f}^{i},\overline{f}^{i}\right]$ (2)

Where $\underline{f}^{i}$ and $\overline{f}^{i}$ can be defined as follow:

$\displaystyle\underline{f}^{i}=\underline{\mu}_{F_{1}^{i}}(x_{1})\ast\ldots% \ast\underline{\mu}_{F_{1}^{n}}(x_{1})$ (3) $\displaystyle\overline{f}^{i}=\overline{\mu}_{F_{1}^{i}}(x_{1})\ast\ldots\ast% \overline{\mu}_{F_{1}^{n}}(x_{1})$ (4)

Figure 2.

Block diagram of proposed FTC scheme [22].

2.2.3 Type reducer

After definition of the rules and executing the inference, the type-2 fuzzy system resulting in type-1 fuzzy system is computed. In this part, the available methods to compute the centroid of type-2 fuzzy system using the extention principle are discussed [27]. The centroid of type-1 fuzzy system A is given by:

$\displaystyle C_{A}=\frac{\sum_{i=1}^{n}z_{i}w_{i}}{\sum_{i=1}^{n}w_{i}}$ (5)

Where: $n$ represents the number of discretized domain of $A$ , $z_{i}\in\mathbb{R}$ and $w_{i}\in[0,1]$ . If each $z_{i}$ and $w_{i}$ is replaced by a type-1 fuzzy system ( $Z_{i}$ and $W_{i}$ ) , with associated membership functions of $\mu_{z}(Z_{i})$ and $\mu_{w}(W_{i})$ respectively, and by using the extention principle, the generalized centroid for type-2 fuzzy system can be expressed by:

$\displaystyle\!\!\!\!\!\!\!\!\!\!\textit{GC}_{\tilde{A}}=\int_{z_{1}\in Z_{1}}% \ldots\int_{z_{n}\in Z_{n}}\int_{w_{1}\in W_{1}}\ldots\int_{w_{n}\in W_{n}}% \left[T_{i=1}^{n}\mu_{Z}(Z_{i})\ast T_{i=1}^{n}\mu_{W}(Z_{i})\right]\Big{/}% \frac{\sum_{i=1}^{n}z_{i}w_{i}}{\sum_{i=1}^{n}w_{i}}$ (6)

$T$ is a $t$ -norm and $\textit{GC}_{\tilde{A}}$ is a type-1 fuzzy system. For an interval type-2 fuzzy system, it can be written:

$\displaystyle\!\!\!\!\!\!\!\!\textit{GC}_{\tilde{A}}=[y_{l}(x),y_{r}(x)]=\int_% {y^{1}\in\left[y_{l}^{1},y_{r}^{1}\right]}\ldots\int_{y^{M}\in\left[y_{l}^{M},% y_{r}^{M}\right]}\ldots\int_{f^{1}\in\left[\underline{f}^{1}\overline{f}^{1}% \right]}\ldots\int_{f^{M}\in\left[\underline{f}^{M}\overline{f}^{M}\right]^{1}% \Big{/}\frac{\sum_{i=1}^{M}f^{i}y^{i}}{\sum_{i=1}^{M}f^{i}}}$ (7)

2.2.4 Defuzzifier

To get a crisp output from a type-1 fuzzy logic system, the type-reduced set must be deffuzzified. The most common method to do this is to find the centroid of the type-reduced set. If the type-reduced set $Y$ is discretized to $n$ points, then the following expression gives the centroid of the type-reduced set:

$\displaystyle y_{\text{output}}(x)=\frac{\sum_{i=1}^{n}y^{i}\mu(y^{i})}{\sum_{% i=1}^{m}\mu(y^{i})}$ (8)

The output can be computed using the iterative Karnik Mendel Algorithm [4, 28, 29, 30, 31]. Therefore, the deffuzzified output of an interval type-2 FLC is:

$\displaystyle y_{\text{output}}(x)=\frac{y_{l}(x)+y_{r}(x)}{2}$ (9)

with:

$\displaystyle\!\!\!\!\!\!\!\!y_{l}(x)=\frac{\sum_{i=1}^{M}f_{l}^{i}y_{l}^{i}}{% \sum_{i=1}^{M}f_{l}^{i}}\text{ and }y_{r}(x)=\frac{\sum_{i=1}^{M}f_{r}^{i}y_{r% }^{i}}{\sum_{i=1}^{M}f_{r}^{i}}$ (10)

2.3 Interval type-2 T-S fuzzy logic controller

2.3.1 Model rule $i$

$\displaystyle\!\!\!\!\!\!\!\!\text{If }e_{1}\text{ is }\tilde{F}_{1}^{i}\text{% and }e_{2}\text{ is }\tilde{F}_{2}^{i}\text{ and }\ldots e_{n}\text{ is }% \tilde{F}_{n}^{i}$

(11) $\displaystyle\!\!\!\!\!\!\!\!\text{then }\dot{x}=A_{i}x(t)+B_{i}u(t),\quad i=1% ,2,\ldots,M$

Here $M$ is the number of IF-THEN rules. $x(t)\in\mathbb{R}^{n\times 1}$ is the state vector of the system. $u(t)\in\mathbb{R}^{m\times 1}$ is the input vector of the system. $A$ is the state matrix of the system. $B$ is the input matrix of the system.
2.3.2 Control rule $i$

$\displaystyle\!\!\!\!\!\!\!\!\text{If }e_{1}\text{ is }\tilde{F}_{1}^{i}\text{% and }e_{2}\text{ is }\tilde{F}_{2}^{i}\text{ and }\ldots e_{n}\text{ is }% \tilde{F}_{n}^{i},$

(12) $\displaystyle\!\!\!\!\!\!\!\!\text{then }\tilde{G}^{i}=1,\ldots,M$

where $\tilde{G}^{i}\in\mathbb{R}^{1\times 1}$ is the output of the interval type-2 fuzzy system for the rule i using the centroid defuzzification.

Figure 3.
Membership functions of input variables $(r(k),r(\dot{k}))$ and output $(u_{k})$ .

2.4 Passive FTC design

In order to eliminate the high oscillation, nonlinear system and model uncertainty, a continuous Interval Type-2 T-S Fuzzy Logic Control (IT2TSFLC) is used to approximate the discontinue control. The proposed control scheme is shown in Fig. 2; it contains conventional PI controller part and IT2TSFLC.

The equivalent control $(u_{\textit{eq}})$ is calculated in such a way to have rate of change of $r(k)$ is zero.

$\displaystyle u_{\textit{fs}}=\textit{IT2TSFLC}(r(k),r(\dot{k}))$ (13)

where $u_{\textit{fs}}$ is the output of the IT2TSFLC, which depends on the normalized $r(k)$ and rate of change of $r(k)$ .

All the membership functions of the fuzzy input variable are chosen to be gaussian for all upper and lower membership functions. The uses labels of the fuzzy variable residue and its derivative are: negative big (NB), negative small (NS), medium (M), Positive small (PS) and positive big (PB). Figure 3 presents the type-2 membership functions for the IT2FLC. The corrective control is decomposed into five levels, so total rules can be 25 presented in Table 1.

$\displaystyle u(k)=u_{fs}+u_{pi}$ (14)

Table 1

Fuzzy rules for IT2FLC

$u_{\textit{fs}}$ & $r(k)$	$\dot{r}(k)$
	NB	NS	M	PS	PB
NB	NB	NS	M	PB	PB
NS	NB	M	M	PB	PB
M	M	M	PS	PS	PB
PS	M	PS	PS	PB	PB
PB	PS	PS	PB	PB	PB

Figure 4.

Two Conical Tank Interacting level System prototype.

3. Process description and mathematical model

The plant used as the test-bench in this work is the Two Conical Tank Interacting Level System (TCTILS), which is shown in Fig. 4. The system contains two conical water tanks that can be filled with two independent pumps acting on the tanks 1 and 2. The liquid flow rate $f_{1}$ and $f_{2}$ are the flow delivered by pump-1 and pump-2. And they can be continuously manipulated from a flow of 0 to a maximum flow $f_{\max}$ .

Figure 5.

Conical Tank Non-interacting level System prototype.

Figure 6.

Regulatory responses for conditions (i and ii) (a) response with system component faults and (b) response with actuator faults.

Interaction in the system is caused by interconnected pipes, which is connected between two conical tanks. The $h_{1}$ and $h_{2}$ are the liquid levels of each tank, it can be measured by differential pressure transmitter (DPT) sensor continuously, and this value can be calibrated in terms of water level [in cm]. The nominal outflow from the system are located at the bottom of tanks, having flow coefficients of $\gamma_{1}$ and $\gamma_{2}$ respectively. By changing the valves in the system enables various configurations, which allows us to work with one, two or three tanks, at a time. The system can be configured as an interacting as well as a non-interacting system. In [32] shown the following dynamic behaviour of the first principles model of the experimental setup:

The mathematical model is derived for both the conical tanks separately as follows.

$\displaystyle A=\pi r^{2}$ (15)

From the Fig. 5

$\displaystyle\tan\theta=\frac{R}{H}$ (16)

At any height $(h_{1})$ of Tank 1

$\displaystyle\tan\theta=\frac{R}{H}=\frac{r}{h_{1}}$ (17)

Table 2

Two Conical Tank Interacting Level System parameters

Sr. No.	Parameter	Value with unit
1	Top radius of the tank $R$	15 cm
2	Height of tank 1 $H_{1}$	60 cm
3	Height of tank 2 $H_{2}$	100 cm
4	Overflow height of tank	120 cm
5	Inner connecting pipe inner diameter	1.25 cm
6	Inner connecting pipe inner diameter	1.25 cm
7	Discharge co-efficient tank 1 $\gamma_{1}$	0.60 cm ${}^{2}$ /sec
8	Discharge co-efficient tank 1 $\gamma_{2}$	0.72 cm ${}^{2}$ /sec
9	Steady state tank 1 $h_{1}$	45 cm
10	Steady state tank 1 $h_{2}$	32 cm
11	Gravitational constant $g$	982 cm ${}^{2}$ /sec

For Tank 1 the mass balance equation is given by following relation [23],

$\displaystyle\frac{dh_{1}}{dt}=\frac{\left[f_{1}-\frac{1}{3}h_{1}\frac{dA(h_{1% })}{dt}-\gamma_{1}\sqrt{h_{1}}\right]}{\frac{1}{3}\pi R^{2}\frac{h_{1}^{2}}{H^% {2}}}$ (18)

Figure 7.

Servo response for conditions (i and ii) (a) response with two abrupt leak faults and (b) zoom part response for Fig. 7a.

Area of the canonical Tank 1 at height $(h_{1})$ and Tank 2 at any height $(h_{2})$

$\displaystyle A_{1}=\frac{\pi R^{2}h_{1}^{2}}{H_{1}^{2}},\quad A_{2}=\frac{\pi R% ^{2}h_{2}^{2}}{H_{2}^{2}}$ (19)

Similarly, for Tank 2 the mass balance equation is given by following relation [23],

$\displaystyle\!\!\!\!\!\!\frac{dh_{2}}{dt}=\frac{\left[f_{2}+\gamma_{1}\sqrt{h% _{1}}-\frac{1}{3}h_{2}\frac{dA(h_{2})}{dt}-\gamma_{2}\sqrt{h_{2}}\right]}{% \frac{1}{3}\pi R^{2}\frac{h_{2}^{2}}{H^{2}}}$ (20)

Where, outlet flow of Tank 1 and Tank 2 given by

$\displaystyle f_{3}=\gamma_{1}\sqrt{h_{1}},\text{ and }f_{o}=\gamma_{2}\sqrt{h% _{2}}$ (21)

The range of the flow rate of pump-1 and pump-2 are about 0 to 253 cm ${}^{3}$ /sec and 296 cm ${}^{3}$ /sec respectively.

The output of the system considering $y=[1\;h_{2}]^{T}$ and input vector $u=[\upsilon_{1}\;\upsilon_{2}]^{T}$ . For the T-S model of the system following matrix is found form the system model:

$\displaystyle\!\!\!\!\!\!\!\!\!\!A_{1}=\begin{bmatrix}0.0215&0.0036\\ 0.0036&0.0215\end{bmatrix},A_{2}=\begin{bmatrix}0.0215&0.0036\\ 0.0036&0.0215\end{bmatrix}$ $\displaystyle\!\!\!\!\!\!\!\!\!\!B_{1}=\begin{bmatrix}54.7321&0\\ 0&54.7321\end{bmatrix},B_{2}=\begin{bmatrix}4.8953&0\\ 0&54.7321\end{bmatrix}$

Figure 8.

Servo response for conditions (i and ii) (a) response with two incipient leak faults and (b) zoom part response for Fig. 8a.

Figure 9.

Servo response for conditions (i and ii) (a) response with two abrupt actuator faults and (b) zoom part response for Fig. 9a.

Figure 10.

Servo response for conditions (i and ii) (a) response with two incipient actuator faults and (b) zoom part response for Fig. 10a.

4. Simulation results

This section presents the results from the simulation of FTC using interval type-2 T-S FLC, T1FLC and PID controller subject to actuator and system component (tank leak) faults and compared with T1FLC.

Table 3
Transient response parameters comparison for IT2TSFLC, T1FLC and PID controller

Transient	Controllers used for fault tolerant control
parameters	IT2TSFLC	T1FLC proposed	PID proposed
		in [33]	in [34]
$T_{r}$	4.126 sec	5.747 sec	5.466 sec
$M_{p}$ (%)	0	0	2.576 %
$T_{s}$	41 sec	68 sec	74 sec

Table 4

IAE and ISE error results from simulation for regulatory performance subject to system component (leak) faults

Controller structure	Abrupt fault		Incipient fault
and condition (i)	IAE	ISE	IAE	ISE
	$h_{2}$	$h_{2}$	$h_{2}$	$h_{2}$
FTC using IT2TSFLC	0.3681	0.6912	0.2581	0.5391
T1FLC proposed in [33]	1.4375	2.5863	1.3872	2.2811
PID proposed in [34]	2.6912	4.9826	2.2814	4.2818

Mainly to assess the performance of FTC with servo and regulatory control, following conditions have been taken:

(i)

When interaction is in effect and system component (tank-2 bottom leak) (Abrupt and Incipient Nature) fault occur.

(ii)

When interaction is in effect and actuator (Abrupt and Incipient Nature) fault occur in $CV_{1}$ and $CV_{2}$ control valve.

Table 5

IAE and ISE error results from simulation for regulatory performance subject to actuator faults

Controller structure	Abrupt fault		Incipient fault
and condition (i)	IAE	ISE	IAE	ISE
	$h_{2}$	$h_{2}$	$h_{2}$	$h_{2}$
FTC using IT2TSFLC	0.6821	0.9712	0.5726	0.8528
T1FLC proposed in [33]	1.9184	3.9528	1.7841	3.6281
PID proposed in [34]	3.1985	5.4383	2.9218	5.0629

Table 6

IAE and ISE error results from simulation for servo performance subject to system component (leak) faults

Controller structure	Abrupt fault		Incipient fault
and condition (i)	IAE	ISE	IAE	ISE
	$h_{2}$	$h_{2}$	$h_{2}$	$h_{2}$
FTC using IT2TSFLC	1.1028	1.8212	1.0025	1.6961
T1FLC proposed in [33]	2.6915	3.2863	2.3925	3.0281
PID proposed in [34]	4.9117	5.6429	4.8594	5.2013

Table 7

IAE and ISE error results from simulation for servo performance subject to system component (leak) faults

Controller structure	Abrupt fault		Incipient fault
and condition (i)	IAE	ISE	IAE	ISE
	$h_{2}$	$h_{2}$	$h_{2}$	$h_{2}$
FTC using IT2TSFLC	1.4368	1.7928	1.2791	1.6691
T1FLC proposed in [33]	2.9261	3.8913	2.8102	3.5289
PID proposed in [34]	5.6819	6.2342	5.5914	6.0941

In this literature, simulation is carried out for condition (i) and (ii) have been accommodated independently.

For the design and simulation of the control system MATLAB environment has been used, in order to validate the efficiency of the propped FTC. The volumetric inflow rate $f_{1}$ and $f_{2}$ of both the pumps pump 1 and pump 2 respectively in the two-tank system used as manipulating variable.

Condition (i) and (ii) of investigation has accommodated together in Fig. 6a. To ensure the possible interaction level and system component faults $(f_{\textit{sys}})$ occur in tank-1 and tank-2 bottom. For simulation two different fault nature ware taken one is abrupt nature and second is incipient nature). Here also same reference-point consider for tank-2 30 cm. Here, first we are letting the controller track the set-point and brings the system in the steady state. After system reaching steady state we ware introducing leak fault in tank-2 bottom. This causes the system to deviate from set-point and degrade the performance, however FTC taking the appropriate action to regulate the level by changing the manipulating variable.

Figure 6b shows the simulation results for regulatory performance of TCTILS for condition-(i and ii). For condition-(i) 30 cm an height has been taken as a reference-point for tank-2, and for condition-(ii) Actuator faults (fa) taken in control valve $\textit{CV}_{1}$ and $\textit{CV}_{2}$ which is controlling the manipulated variable $f_{1}$ and $f_{2}$ volumetric inflow rate of pump 1 and pump 2 of TCTILS hybrid system. In this simulation the actuator faults (fa) behavior is abrupt and incipient in nature applied at $t=$ 83 sec after steady state height is achieved. Control signal generated by IT2TSFLC and PI controller for changing, manipulating variable (which is inflow rate) has been plotted in Fig. 6b, in order to represent the performance of controller with time. The two conventional controller (i.e. PID and T1FLC controller) scheme is also implemented to compare with proposed IT2TSFLC scheme. In Table 3, transient performance presented for three control scheme for FTC, which clearly demonstrates among three scheme IT2TSFLC are gives quick response, zero overshoot and almost zero steady state error. Form the simulation error value are obtained, as the minimal value of ISE will represent the elimination of large errors quickly and often this leads to faster responses. The IAE and ISE error results for proposed FTC scheme presented in Tables 4 and 5 subject to different conditions for regulatory performance.

5. Conclusion

In this paper, efficient FTC scheme is proposed using an Interval Type-2 T-S FLC with PI controller for a Two Conical Tank Interacting Level System (TCTILS) subject to actuator and system component faults, the simulation results have been produced and efficacy of the proposed FTC investigated using IAE and ISE error. Also the proposed scheme is compared with conventional PI and T1FLC controller, transient parameters verifies the efficacy of the proposed scheme as compared to others. The key benefit of the IT2TSFLC appears to be its expertise to remove persistent fluctuations, model uncertainty and nonlinear behavior of the system. As the IT2TSFLC can tolerate larger faults and no need to measured accurate faults magnitudes. The prime benefits of the proposed strategy is to tolerate system component and actuator faults and maintain control performance at acceptable level.

References

Patel

H.R.

and Shah

V.A.

, Fault tolerant control systems: A passive approaches for single tank level control system, i-managers, Journal on Instrumentation and Control Engineering 6(1) (2018), 11–18, doi: https://doi.org/10.26634/jic.6.1.13934.

Patel

H.R.

and Shah

V.A.

, Fuzzy logic based passive fault tolerant control strategy for a single-tank system with system fault and process disturbances, in: IEEE, 5th International Conference on Electrical and Electronic Engineering (ICEEE 2018), Istanbul, Turkey, 2018, pp. 257–262. doi: 10.1109/ICEEE2.2018.8391342.

Hao

Bin

and Marcel

, Fault recoverability and fault tolerant control for a class of interconnected non-linear systems, Automatica 54 (2015), 49–55, doi: https://doi.org/10.1016/j.automatica.2015.01.037.

Castillo

and Melin

, A review on the design and optimization of interval type-2 fuzzy controllers, Application of Soft Computing 12 (2012), 1267–1278, doi: https://doi.org/10.1016/j.asoc.2011.12.010.

Guerra

Puig

and Witczak

, Robust fault detection with unknown input set-membership state estimators and interval models using zonotopes, in: Proceedings of the 6th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, SAFEPROCESS 2006, Beijing, China, 2006, pp. 1303–1308.

Patel

H.R.

and Shah

V.A.

, Fault detection and diagnosis methods in power generation plants-the Indian power generation sector perspective: An introductory review, Journal of Energy and Management 2(2) (2018), 31–49.

Patton

, Fault-tolerant control systems: The 1997 situation, in: Proceedings of the IFAC Symposium on Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, 1997, 1033–1054.

Korbicz

Kocielny

Kowalczuk

and Choleva

, Fault Diagnosis: Models, Artificial Intelligence, Applications, Springer-Verlag, Berlin/Heidelberg, New York, NY, 2004, pp. 920.

Zhang

and Jiang

, Bibliographical review on reconfigurable fault-tolerant control systems, Annual Reviews in Control 32(2) (2008), 229–252, doi: https://doi.org/10.1016/j.arcontrol.2008.03.008.

10.

Noura

Theilliol

Ponsart

and Chamssedine

, Fault-Tolerant Control Systems: Design and Practical Applications, in: Advances in Industrial Control, Springer-Verlag, London, 2009, doi: 10.1007/978-1-84882-653-3.

11.

Puig

, Fault diagnosis and fault tolerant control using set-membership approaches: Application to real case studies, International Journal of Applied Mathematics and Computer Science 20(4) (2010), 619–635, doi: 10.2478/v10006-010-0046-y.

12.

Hao

Marcel

Bin

and Jianye

, Fault tolerant cooperative control for a class of nonlinear multi-agent systems, Systems and Control Letters 604 (2011), 271–277, doi: https://doi.org/10.1016/j.sysconle.2011.02.004.

13.

Y.Hao Bin

Marcel

and Youmin

, Fault recoverability and fault tolerant control for a class of interconnected nonlinear systems, Automatica 54 (2015), 49–55, doi: https://doi.org/10.1016/j.automatica.2015.01.037.

14.

Hupo

and Yan

, Adaptive fault-tolerant control for actuator failures: A switching strategy, Automatica 81 (2017), 87–95, doi: https://doi.org/10.1016/j.automatica.2017.03.014.

15.

Boskovic

J.D.

and Mehra

R.K.

, Failure detection, identification and reconfiguration in flight control, in: Fault Diagnosis and Fault Tolerance for Mechatronic Systems, Springer-Verlag, New York, 2002, doi: https://doi.org/10.1007/3-540-45737-25.

16.

Castillo

and Melin

, Type-2 Fuzzy Logic: Theory and Applications, Springer Verlag, Germany, 2008.

17.

Wang

, Interval type-2 fuzzy T-S modeling for a heat exchange process on CE117 Process Trainer, in: Proceedings of 2011 International Conference on Modelling, Identification and Control, 2011, pp. 457–462.

18.

Sudha

K.R.

and Santhi

R.V.

, Robust decentralized load frequency control of interconnected power system with Generation Rate Constraint using Type-2 fuzzy approach, Department of Electrical Engineering 33 (2011), 699–707.

19.

Lam

H.K.

and Seneviratne

L.D.

, Stability analysis of interval type-2 fuzzy-model-based control systems, IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics 389(3) (2008), 617–628, doi: 10.1109/TSMCB.2008.915530.

20.

Patel

H.R.

and Shah

V.A.

, A framework for fault-tolerant control for an interacting and non-interacting level control system using AI, in: Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics-Volume 1: ICINCO, ISBN 978-989-758-321-6, 2018, pp. 180–190. doi: 10.5220/0006862001800190.

21.

Patel

H.R.

and Shah

V.A.

, A fault-tolerant control strategy for non-linear system: An application to the two tank canonical non-interacting level control system, in: Proceedings of 2nd International Conference on Distributed Computing, VLSI, Electrical Circuits and Robotics (DISCOVER 2018), Mangalore, India, 13–14 August 2018.

22.

Patel

H.R.

and Shah

V.A.

, Fault tolerant control using interval type-2 takagi-sugeno fuzzy controller for nonlinear system, in: Proceedings of 18th International Conference on Intelligent Systems Design and Applications (ISDA-2018), VIT university, Vellore, India, 6–8 December, 2018.

23.

Patel

H.R.

and Shah

V.A.

, A passive fault-tolerant control strategy for non-linear system: An application to the two tank conical non-interacting level control system, MASKAY Journal 9(1) (2019), 1–8.

24.

Zadeh

L.A.

, The concept of a linguistic variable and its application to approximate reasoning-1, Information Sciences 8(3) (1975), 199–249, doi: https://doi.org/10.1016/0020-0255(75)90036-5.

25.

Karnik

and Mendel

, Introduction to type-2 fuzzy logic systems, in: Fuzzy Systems Proceedings IEEE World Congress on Computational Intelligence, Anchorage, AK, USA, 2, 1998, pp. 915–920, doi: 10.1109/FUZZY.1998.686240.

26.

Liang

and Mendel

, Interval type-2 fuzzy logic systems: Theory and design, IEEE Transactions on Fuzzy Systems 8(5) (2000), 535–550, doi: 10.1109/91.873577.

27.

Mendel

, Uncertain Rule-based Fuzzy Logic Systems: Introduction and New Directions, Prentice-Hall, 2011, doi: 10.1007/978-3-319-51370-6.

28.

Juan

Castillo

Melin

and Rodrguez-Daz

, A hybrid learning algorithm for a class of interval type-2 fuzzy neural networks, Information Sciences 179(13) (2009), 2175–2193, doi: https://doi.org/10.1016/j.ins.2008.10.016.

29.

Martnez

Castillo

and Aguilar

, Optimization of interval type-2 fuzzy logic controllers for a perturbed autonomous wheeled mobile robot using genetic algorithms, Information Sciences 179(13) (2009), 2158–2174, doi: https://doi.org/10.1016/j.ins.2008.12.028.

30.

and Tan

, A simplified type-2 fuzzy logic controller for real-time control, ISA Transactions 45(4) (2006), 503–516, doi: https://doi.org/10.1016/S0019-0578(07)60228-6.

31.

Mndez

Martinez

Gonzlez

and Rendn-Espinoza

, Orthogonal-least-squares and back propagation hybrid learning algorithm for interval A2-C1 singleton type-2 Takagi-Sugeno-Kang fuzzy logic systems, International Journal of Hybrid Intelligent Systems 11(2) (2014), 125–135, doi: 10.3233/HIS-130188.

32.

Nandola

and Bhartiya

, A multiple model approach for the predictive control of nonlinear hybrid system, Journal of Process Control 18(2) (2008), 131–148, doi: https://doi.org/10.1016/j.jprocont.2007.07.003.

33.

Qingsong

Wang

and Hongling

, Design of fuzzy controller for liquid level control system based on MATLAB/RTW, in: Proceedings of the 2nd IEEE International Conference on Measurement, Information and Control, Vol. 2, Harbin, China, 2013, pp. 1090–1094, doi: 10.1109/MIC.2013.6758149.

34.

Holi

and Vesel

, Robust PID controller design for coupled-tank process using Labreg software, IFAC Proceedings Volumes 45(11) (2012), 442–447, doi: https://doi.org/10.3182/20120619-3-RU-2024.00100.

$u_{\textit{fs}}$ & $r(k)$	$\dot{r}(k)$
	NB	NS	M	PS	PB
NB	NB	NS	M	PB	PB
NS	NB	M	M	PB	PB
M	M	M	PS	PS	PB
PS	M	PS	PS	PB	PB
PB	PS	PS	PB	PB	PB

$u_{\textit{fs}}$ & $r(k)$	$\dot{r}(k)$
	NB	NS	M	PS	PB
NB	NB	NS	M	PB	PB
NS	NB	M	M	PB	PB
M	M	M	PS	PS	PB
PS	M	PS	PS	PB	PB
PB	PS	PS	PB	PB	PB

Actuator and system component fault tolerant control using interval type-2 Takagi-Sugeno fuzzy controller for hybrid nonlinear process

Abstract

Keywords

1. Introduction

2.1 Background of the type-2 fuzzy logic control

2.2 Type-2 fuzzy logic control

2.2.1 Fuzzifier

2.3.1 Model rule i

Table 3 Transient response parameters comparison for IT2TSFLC, T1FLC and PID controller

References

2.3.1 Model rule $i$

Table 3
Transient response parameters comparison for IT2TSFLC, T1FLC and PID controller

$u_{\textit{fs}}$ & $r(k)$	$\dot{r}(k)$
	NB	NS	M	PS	PB
NB	NB	NS	M	PB	PB
NS	NB	M	M	PB	PB
M	M	M	PS	PS	PB
PS	M	PS	PS	PB	PB
PB	PS	PS	PB	PB	PB