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Probabilistic fuzzy logic controller for uncertain nonlinear systems
Accepted Manuscript
Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems Omar Shaheen , Ahmad M. El-Nagar , Mohammad El-Bardini , Nabila M. El-Rabaie PII: DOI: Reference:
S0016-0032(18)30023-1 10.1016/j.jfranklin.2017.12.015 FI 3267
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
22 March 2017 30 October 2017 4 December 2017
Please cite this article as: Omar Shaheen , Ahmad M. El-Nagar , Mohammad El-Bardini , Nabila M. El-Rabaie , Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2017.12.015
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ACCEPTED MANUSCRIPT
Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems
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Omar Shaheen1, Ahmad M. El-Nagar2, Mohammad El-Bardini3 and Nabila M. El-Rabaie4 Department of Industrial Electronics and Control Engineering Faculty of Electronic Engineering, Menoufia University Menouf, 32852, Egypt 1 [email protected] 2 [email protected] 3 [email protected] 4 [email protected]
Abstract:
This paper proposes a probabilistic fuzzy proportional - integral (PFPI) controller for controlling
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uncertain nonlinear systems. Firstly, the probabilistic fuzzy logic system (PFLS) improves the capability of the ordinary fuzzy logic system (FLS) to overcome various uncertainties in the controlled dynamical systems by integrating the probability method into the fuzzy logic system. Moreover, the input/output relationship for the proposed PFPI controller is derived. The resulting structure is equivalent to nonlinear PI controller and the equivalent gains for the proposed PFPI controller are a
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nonlinear function of input variables. These gains are changed as the input variables changed. The sufficient conditions for the proposed PFPI controller, which achieve the bounded-input bounded-
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output (BIBO) stability are obtained based on the small gain theorem. Finally, the obtained results indicate that the PFPI controller is able to reduce the effect of the system uncertainties compared with
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the fuzzy PI (FPI) controller. Keywords
Fuzzy logic controller, Probabilistic fuzzy logic system, stochastic uncertainties, Probabilistic fuzzy
1. Introduction
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controller, Analytical structure, BIBO stability
Most practical engineering systems are complex and time-variant with nonlinearity. These systems
contain a time-delay, an external disturbance and a modeling error. These problems reduce the performance of the systems [1, 2]. It is undesirable to be worthily controlled these systems using the classical controllers [3]. The design of modern controllers such as adaptive controllers contains complex mathematical analysis and also, have many difficulties in controlling time-varying and highly nonlinear plants [4]. For these reasons, many researchers have attempted to use intelligent controllers such as fuzzy control and neural network control in order to improve the system performance [4 - 7]. 1
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The presence of the system uncertainties increases the merits of the use of the fuzzy control as compared to other controllers due to the capability of fuzzy reasoning to reduce the effects of system uncertainties [8 - 13]. There are exist different uncertainties accompanied with the practical processes. These uncertainties may be stochastic and non-stochastic uncertainties [14]. The uncertainties due to the plant dynamics can be represented by non-stochastic uncertainties, but the uncertainties due to random noise
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in the measured data can be considered as stochastic uncertainties [15, 16]. Practically, most real world applications contain both stochastic and non-stochastic uncertainties and it is difficult to control such systems due to increasing uncertainties. However, the ordinary FLC (Type-1 fuzzy logic controller; T1FLC), which uses the ordinary fuzzy sets could not handle stochastic uncertainty and consequently, the T1FLC is not the effective controller for controlling the processes with stochastic and non-
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stochastic uncertainties [15].
The fuzzy logic system and the probability theory have been combined together in a valuable strategy to process uncertainties [17, 18]. The probabilistic fuzzy logic system (PFLS) is introduced for reducing the effect of stochastic and non-stochastic uncertainties. The PFLS uses the probabilistic
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fuzzy set (PFS) that is characterized via a three-dimensional membership function for capturing stochastic feature. The additional stochastic expression describes the secondary probability density
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function of the random membership grades [8, 19 - 21]. Therefore, the PFS provides more degrees of freedom that make the PFLS is an effective tool to reduce the effect of system uncertainties. Similar to FLSs, the PFLS contains three operations, which are a fuzzification process, an inference engine and a
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defuzzification operation [15]. However, in the PFLS, the reasoning mechanism is integrated with the probabilistic method. The PFLS was proposed and applied for system modeling [15, 19, 22 - 24],
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control of the process with stochastic uncertainties [15, 25 - 27] and function approximation [28]. Most of the researchers focused on the PFLS for system modeling. However, the research about PFLS for
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controlling the uncertain systems still exists. To date, there is no analytical structure and stability analysis for the PFLS to show the effect of the parameters of the probability density function on the system performance. The main purpose of this paper is developing the PFPI controller taking into its consideration the
advantages of the PFS to overcome the stochastic and non-stochastic uncertainties. This paper attempts to discuss the analytical structure of the proposed PFPI controller, which is an efficient method to show the input/output relationship for the proposed controller [29, 30]. Deriving the analytical structure of 2
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the probabilistic fuzzy logic controller (PFLC) is complicated than the T1FLC where the PFLC contains more degree of freedom and more complicated inference engine. The sufficient conditions, which achieve the BIBO stability for the proposed PFPI controller, are derived based on the small gain theorem. The proposed controller is applied for controlling two uncertain nonlinear systems. These are nonlinear mathematical plant and the continuous stirred tank reactor (CSTR) process. Simulation results indicate that the PFPI controller is a robust controller, which able to handle various uncertainties
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in the controlled system.
Our study has several contributions, which summarized as: (1) Probabilistic fuzzy PI controller using probabilistic fuzzy set is proposed to control uncertain nonlinear systems. (2) Deriving the analytical structure for the proposed PFPI controller to obtain the input/output relationship for the
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proposed controller. (3) Driving the stability conditions of the PFPI controller.
The rest of this paper is organized as follows: The structure of the PFPI controller is introduced in section 2. The analytical structure of the proposed PFPI controller is introduced in section 3. The stability analysis of the PFPI controller is presented in section 4. The simulation results are presented in section 5. Finally, the conclusions and the relevant references are presented.
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2. Probabilistic Fuzzy PI Controller
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The block diagram, which describes the main components of the proposed PFPI controller is illustrated in Fig. 1. It consists of three input/output variables. The two inputs are the error ( e(k ) ) and
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the change of error ( v(k ) ) and the output is the change of control signal ( U PFPI (k ) ).
e(k )
en (k )
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Ge
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Probabilistic Fuzzy Logic System
v(k )
Gv
vn (k )
Primary Fuzzification
Fuzzy Rules Traditional Defuzzification
Mamdani Inference
Probabilistic Processing
Bayesian Inferenc e
Probability Calculation Probabilistic Fuzzifcation
Probabilistic Defuzzifcation
Inference
Fig.1: Main components of the PFPI controller.
The two input variables have been scaled as follows: 3
U (k )
Gu
u (k )
uPFPI (k ) z 1
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en (k ) Ge (r (k ) ym (k ))
(1)
vn (k ) Gv (e(k ) e(k 1))
(2)
where r (k ) is the reference input, ym (k ) is the measured output, and k is the sampling instance. Ge and Gv are the scaling factors for the input variables e(k ) and v(k ) , respectively. The control signal
uPFPI (k ) is given as:
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uPFPI (k ) uPFPI (k 1) Gu U PFPI (k )
(3)
where uPFPI (k 1) is the previous value of uPFPI (k ) , U PFPI (k ) is the change in the denormalized output, and Gu is the output scaling factor.
The processing configuration of the PFPI controller consists of four major operations: the probabilistic fuzzification, rule base, the probabilistic fuzzy inference engine and the probabilistic
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defuzzification. Briefly, the proposed controller works as follows: In the probabilistic fuzzification process, the crisp input is fuzzified and the membership grade can be obtained by the primary fuzzification. The randomness uncertainties of the input data can be handled through probabilistic calculation. Then, the probabilistic fuzzy inference engine is used to infer the probabilistic fuzzy output
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based on the fuzzy rule-based. Finally, the crisp output can be obtained using the probabilistic defuzzification that consists of two parts; an ordinary defuzzification and probabilistic processing.
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2.1 Probabilistic fuzzification
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~ The PFS A can be expressed by the probability space ( x , , P) and it is represented in Fig. 2 [15]. ~ A ( x , , P) x X
(4)
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where x is the set of all possible events, is the field and P is the probability defined on . For a crisp input, the membership grade in the ordinary fuzzification is a single value. However, in
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the probabilistic fuzzification, it is a random variable with a certain probabilistic distribution function (PDF). The probabilistic fuzzification consists of two parts; the first part is the primary fuzzification, which obtains the membership grades ( x)0,1 of the crisp inputs in the antecedent part of the rules and the probabilistic fuzzification for calculating the corresponding probability of all possible membership grades for the controller inputs. In the probability calculation process, the Bayes' theorem processes the probabilistic information taking into account the group of fuzzy grades being independent variables. The probability function 4
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that is used for calculating the corresponding probability of all possible fuzzy membership grades is a Gaussian function. The probability can be calculated as: ( mi ) 2
Pi ( ) e
i2
(5)
where mi and i are the mean and standard deviation of the Gaussian function, respectively. To reduce the computation load, the fuzzy membership grades are divided into three cells i 1,2,3 and
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the weights mi and i are all set to constants. In this paper, i is set to be 1 [26, 27]. The joint probability of the membership grade is calculated as the following based on the Bayes' probability theorem: 3
P( ) Pi ( )
(6)
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i 1
P( )
(x) Primary MF
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1
Probabilistic density function
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P( )
(x)
(x) x
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Crisp input
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Fig.2: Probabilistic Fuzzy Set (PFS).
In this paper, the controller inputs en (k ) and vn (k ) are represented by two PFSs. The primary
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fuzzy sets are described by two triangle membership function called “positive” and “negative”, where the primary membership function of the controller inputs are described in Eqs. (7, 8) in which the constants q1 and q2 are design parameters.
e P n
(en (k ) q1 ) / 2q1 1 0
q1 en (k ) q1 en (k ) q1
e N n
otherwise
5
1 (q1 en (k )) / 2q1 0
en (k ) q1 q1 en (k ) q1 other wis e
(7)
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v P n
(vn (k ) q2 ) / 2q2 1 0
q2 vn (k ) q2
VN
vn (k ) q2 otherwise
1 (q2 vn (k )) / 2q2 0
vn (k ) q2 q2 vn (k ) q2
(8)
otherwise
2.2 Rule-based The universe of discourse for the controller output U PFPI (k ) is divided into three overlapping
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PFSs where the primary fuzzy set is represented by triangular membership function labeled N (Negative), Z (Zero) and P (Positive). The rule-based for the proposed PFPI controller is described in Table 1.
Table 1: Rule-based for the proposed PFPI controller.
Derivative of error signal P N
Error Signal
N
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P P Z
2.3 Probabilistic fuzzy inference
Z N
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The inference engine produces fuzzy output based on the rule-based. The probabilistic frameworks are used to perform the probabilistic fuzzy inference. The AND operations in the fuzzy rule-based are
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represented by the minimum operation (min ()). Thus, the firing grades, which are the result of the minimum operator for the four fuzzy rules are obtained as:
R~1 min( e P , v P ) for U PI (k ) P
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n
n
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R~ 2 min( e P , v N ) for U PI (k ) Z n
n
(9) (10)
R~3 min( e N , v P ) for U PI (k ) Z
(11)
R~ 4 min( e N , v N ) for U PI (k ) N
(12)
n
n
n
n
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The probability of the firing grade R~j can be obtained as in Eq. (13) based on the Bayes’
inference method:
P( R~j ) P(~en j ) P(v~n j )
(13)
where P( R~j ) , P( e~n j ) and P( v~n j ) denote the corresponding probability of the firing grade R~j , ~en j and v~n j ; with considering ~en j and v~n j as independent events. 6
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2.4 Probabilistic defuzzification This operation is used to obtain the crisp output based on the probabilistic processing method. It uses the PFSs instead of the ordinary fuzzy sets in the output of the proposed PFPI controller. The output domain of the PFS is divided into N points and the output of the centroid defuzzification can be obtained as:
y
l 1 N
l
R ( x, y l )
l 1
R
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N
y
( x, y l )
(14)
where y and R are random variables. To obtain the crisp output for the proposed PFPI controller, we perform the mathematical expectation on Eq. (14) as:
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U PFPI (k ) EX ( y)
(15)
One of the defuzzification approaches is concerned with calculation the mathematical expectation and the crisp output is obtained as:
N
l 1 N
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U PFPI (k )
C E ( l
E ( l 1
R
R
)
)
C N E ( N ) CZ E ( Z ) C P E ( P ) E ( N ) E (Z ) E (P )
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U PFPI (k )
(16)
(17)
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where C N , CZ and C P are the centroid for three output fuzzy sets. The mathematical expectation of
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the fuzzy output variables can be obtained as: E (P ) R~1 P(R~1 )
E ( Z ) R~ 2 P( R~ 2 ) R~3 P( R~3 ) E (N ) R~ 4 P(R~ 4 )
(18) (19) (20)
Assume CZ 0 , then: U PI (k )
C N E ( N ) CP E ( P ) E( N ) E(Z ) E( P )
7
(21)
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3. Analytical Structure of the Probabilistic Fuzzy PI Controller In this section, we present the analytical structure for the proposed PFPI controller. The min() operators in the four fuzzy rule-based, which defined in Eqs. (9-12) are performed to obtain the mathematical input/output relationship of the proposed PFPI controller. Assume that en (k ) is inside
[q1 , q1 ] and vn (k ) is inside [q2 , q2 ] for brevity. The final values for the four fuzzy rules in Eqs. (9-12)
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and their corresponding probabilities are first calculated to derive the proposed controller output, which defined in Eq. (21). After that, we can calculate the mathematical expectation of the fuzzy output variables from Eqs. (18–20) and finally, the crisp output of the proposed PFPI controller can be obtained. To obtain which membership value is smaller in each fuzzy rule, the input space
[q1 , q1 ] [q2 , q2 ] is divided into two regions in each rule, each region is called input combination
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(IC). However, all the four rules are executed at the same time. We obtain a total 4 ICs by superimposing all rules as shown in Fig. 3. Table 2, indicates the result of the firing grades and their corresponding probabilities for these 4 ICs.
vn P
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vn N
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q2
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vn (k )
2
3
1 4
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q2 en N
en P
q1
q1
Fig. 3. Input space division.
8
en (k )
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Table 2: The firing grades and their corresponding probabilities of the four rules for IC1 to IC4.
Rule 1
R~1
R~ 3
P( R~ 3 )
R~ 4
P( R~ 4 )
v N
e g2
e N
e g3
e N
e g4
e P
e g1
v N
e g2
e N
e g3
v N
e g4
e P
e g1
e P
e g2
v P
e g3
v N
e g4
v P
e g1
e P
e g2
v P
e g3
e N
e g4
n
n
4
P( R~ 2 )
e g1
n
3
R~ 2
Rule 4
v P
1 2
P( R~1 )
Rule 3
n
n
n
n
n
where the parameters g1, g2 , g3 and g 4 are defined as:
n
n
n
n
n
n
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IC No.
Rule 2
n
n
g1 3 Pen (2m1 2m2 2m3 ) Pen 3 Pvn (2m1 2m2 2m3 ) Pvn (2m12 2m22 2m32 ) 2
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2
g 2 3 Pen (2m1 2m2 2m3 ) Pen 3 Nvn (2m1 2m2 2m3 ) Nvn (2m12 2m22 2m32 ) 2
2
(22) (23)
g 3 3 Nen (2m1 2m2 2m3 ) Nen 3 Pvn (2m1 2m2 2m3 ) Pvn (2m12 2m22 2m32 )
(24)
g 4 3 Nen (2m1 2m2 2m3 ) Nen 3 Nvn (2m1 2m2 2m3 ) Nvn (2m12 2m22 2m32 )
(25)
2
2
2
2
For each IC, the mathematical expectation is calculated from Eqs. (18-20) and then, the outputs of
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these equations are used in Eq. (21) to obtain the output of the controller. After some mathematical operations, the mathematical expressions of the proposed PFPI controller are obtained for each IC.
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Finally, the mathematical expressions of u PI (k ) for all the ICs that shown in Fig. 3 share the same
C1r Ge e(k ) C2r Gv v(k ) C3r D1r Ge e(k ) D2r Gv v(k ) D3r
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structure pattern as: r u PFPI (k ) Gu
r 1,...,4
(26)
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where Cn and Dn , for n 1,2,3 are constants. The values of these constant depend on the parameters of the proposed PFPI controller.
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Letting
K pr (e(k ), v(k ))
Gu C2r Gv D1r Ge e(k ) D2r Gv v(k ) D3r
K ir (e(k ), v(k ))
Gu C1r Ge D1r Ge e(k ) D2r Gv v(k ) D3r
Gu C3r (e(k ), v(k )) r D1 Ge e(k ) D2r Gv v(k ) D3r r
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Then r uPFPI (k ) K pr (e(k ), v(k )) v(k ) Kir (e(k ), v(k )) e(k ) r (e(k ), v(k ))
(27)
In order to simplify the analysis, the parameters of the proposed PFPI are assumed as:
q1 q2 q , CP R, CN R where R 0 . Tables 3 and 4 indicate the values of the proportional and integral gains for the proposed PFPI controller in each IC, respectively. The proportional and integral
uFPPI (k ) ( K p Ki ) e(k ) K p e(k 1)
Letting
K1P (e(k ), v(k )) ( K p Ki ) , and K 2P (e(k ), v(k )) K p
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gains are a nonlinear function of controller parameters. Equation (27) can be rewritten in general as:
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uPFPI (k ) K1P (e(k ), v(k )) e(k ) K 2P (e(k ), v(k )) e(k 1) (e(k ), v(k ))
(28)
(29)
Equation (30) is defined the digital form of the linear PI controller as:
uPId (k ) uPId (k ) uPId (k 1) K1d e(k ) K 2d e(k 1)
(30)
where K1d and K 2d are the digital linear PI controller parameters. From Eq. (29) and Eq. (30), we find that the proposed PFPI controller is a nonlinear PI controller in an incremental form with the variable
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gains K1P (e(k ), v(k )) , K 2P (e(k ), v(k )) and the variable control offset (e(k ), v(k )) . Remark 1: The PFPI controller has six control parameters free to design ( Ge , Gv , GU , m1 , m2 , m3 ),
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whereas the FPI controller only has three parameters ( Ge , Gv , GU ). If these parameters are used in the designing of the proposed PFPI controller, we have more degree of freedom that makes the proposed
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controller are able to improve the system performance and reduce the effect of system uncertainties. Table 3: The dynamic proportional gain for the PFPI controller.
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IC No.
Kp
1
R Gv Gu e g1 (e g3 e g4 )Ge e (e g1 e g2 )Gv v q(e g1 e g2 e g3 e g4 )
2
R Gv Gu e g4 (e g1 e g3 )Ge e (e g2 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
3
R Gv Gu e g4 (e g1 e g2 )Ge e (e g3 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
4
RGv Gu e g1 (e g2 e g4 )Ge e (e g1 e g3 )Gv v q(e g1 e g2 e g3 e g4 )
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Table 4: The dynamic integral gain gains for the PFPI controller.
IC No.
Ki
2
R Ge Gu e g1 (e g1 e g3 )Ge e (e g2 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
3
R Ge Gu e g1 (e g1 e g2 )Ge e (e g3 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
4
R Ge Gu e g4 (e g2 e g4 )Ge e (e g1 e g3 )Gv v q(e g1 e g2 e g3 e g4 )
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1
R Ge Gu e g4 (e g3 e g4 )Ge e (e g1 e g2 )Gv v q(e g1 e g2 e g3 e g4 )
4. Stability Analysis
The small gain theorem is used in this section to drive the sufficient conditions for the proposed PFPI controller to achieve the BIBO stability [31]. Fig. 4, shows the feedback system where the proposed PFPI controller is represented by subsystem S1 while the subsystem S 2 represents the
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controlled system. The following equations describe the overall feedback system:
u1
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e1 u1 y2 , e2 u2 y1 , y1 S1 e1 and y2 S 2 e2 . e1
+
y1
S1
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-
e2
y2
S2
+ +
u2
Fig. 4: Feedback control system.
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Assume the gain of S1 is 1 (S1 ) , and the gain of S 2 is 2 (S 2 ) . Assume that there are constants
1, 2 , 1 0 and 2 0 so that: y1 S1e1 1 e1 1
(31)
y2 S 2e2 2 e2 2
(32)
Based on the small gain theorem, the sufficient condition, which makes the controlled system is BIBO stable is 1 2 1 . A bounded output pair ( y1, y2 ) is produced at any bounded input pair (u1 , u2 ) . 11
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Let we assume the controlled system is nonlinear described by N . The feedback system that shown in Fig. 4 is obtained by defining r (k ) u1 (k ) , e(k ) e1 (k ) , uPFPI (k ) y1 (k ) , uPI (k 1) u2 (k ) ,
uPFPI (k ) e2 (k ) , y(k ) y2 (k ) in Fig. 1. The sufficient conditions for the proposed PFPI controller, which achieve the BIBO stability for each IC is derived as the following: According to Eq. (29), we obtain the following equation as:
where M e is the maximum magnitude of e(k ) that described as: M e : sup e(k ) k 0
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uPFPI (k ) y1 (k ) H1e1 (k ) K1 e1 (k ) K 2 M e
(33)
(34)
According to Eq. (31) and Eq. (33), we have 1 K1 . Then, the following equation is obtained as:
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y(k ) y2 (k ) H 2e2 (k ) Ne2 (k ) N e2 (k )
(35)
By comparing Eq. (35) with Eq. (32), we have 2 N . Therefore, the controlled system based on the proposed PFPI controller is BIBO stable if the parameters of the proposed controller must satisfy
defined as:
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the inequality K1 N 1 . Where K1 is defined for the regions IC1 to IC4 as in Table 5, where M v is
M v : sup v(k ) sup v(k ) v(k 1)
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k 0
Table 5: Values of
K1
RGu ( Ge e g4 Gv e g1 ) (e g3 e g4 )Ge M e (e g1 e g2 )Gv M v q(e g1 e g2 e g3 e g4 )
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1
K1 for stability conditions of the PFPI controller.
PT
IC No.
(36)
k 0
2
RGu ( Ge e g1 Gv e g4 ) (e g1 e g3 )Ge M e (e g2 e g4 )Gv M v q(e g1 e g2 e g3 e g4 )
3
RGu ( Ge e g1 Gv e g4 ) (e g1 e g2 )Ge M e (e g3 e g4 )Gv M v q(e g1 e g2 e g3 e g4 )
4
RGu ( Ge e g4 Gv e g1 ) (e g2 e g4 )Ge M e (e g1 e g3 )Gv M v q(e g1 e g2 e g3 e g4 )
The sufficient condition for the proposed PFPI controller, which achieve the BIBO stability is obtained by combining together all the conditions that shown in Table 5. 12
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Remark 2: The sufficient condition of the BIBO stability for the proposed PFPI controller is dependent on the system parameters ( Ge , Gv , GU , m1 , m2 , m3 ). Therefore, the values of these parameters must be chosen to make the system gains achieve the condition of the BIBO stability.
4. Simulation Results To show the robustness of the PFPI controller, we present two nonlinear systems. The simulation
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results of the proposed PFPI controller are compared with that of the FPI controller and PFLS, which is proposed previously in [21]. The comparison between two controllers is based on the same number of the membership functions, the number of rules and the same scaling factors. The control system performance is compared in terms of the root-mean-squared errors (RMSE), the mean absolute errors (MAE) and the mean absolute percentage errors (MAPE), which are given as:
MAE
1 N
y (k ) y (k ) k 1
k 1
(37)
N
1 N N
2
d
y (k ) y (k )
(38)
yd (k ) y(k ) 100 % yd (k )
(39)
k 1
d
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MAPE
N
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1 N
RMSE
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Example 1: This example demonstrates the performance of the proposed PFPI controller for a nonlinear system. Consider the nonlinear system is defined by the following:
(a1 a1 ) y p (k 1) y p (k 2) y p (k 1) (a2 a2 )
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y p (k )
(a3 a3 ) y p (k 1) 2 y p (k 2) 2
(b b)u(k 1)
(40)
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where the normal values of the system parameters are a1 1, a2 1.5, a3 1and b 1 . a1 , a2 , a3 and
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b are the uncertainty values due to the uncertainty in the parameters of the system. y p (k ) and u (k )
are the nonlinear system output and the control signal, respectively. Simulation tasks are performed with sampling time 0.1 sec. Several simulation tasks are performed to show the robustness of the proposed PFPI controller. The set-point is a unit step input for all the simulation tasks. Task 1: This simulation task indicates the performance of the PFPI controller under the effect of the uncertainty in system’s parameters. There are two cases as the following:
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Case
1:
The
uncertainties
in
the
system’s
parameters
with
the
values:
(a1 0.13, a2 0.13, a3 0.13, b 0.13) are added at time equal 100 sec. The response of the
controlled system for the proposed PFPI, PFLS and FPI controllers is shown in Fig.5. The proposed PFPI controller can provide a robust performance after adding the uncertainty values. The performance of the PFPI controller has smaller overshoot and settling time than that obtained for PFLS and FPI
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controller.
Fig. 5. Response of the nonlinear system (Task 1 - case 1). 2:
The
uncertainties
in
the
system’s
parameters
with
the
values:
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Case
(a1 0.14, a2 0.14, a3 0.14, b 0.14) are added at time equal 100 sec. The system response
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for this case is shown in Fig.6. it’ clear that the controlled system remains stable after adding the uncertainty values with the proposed PFPI controller but it becomes unstable with the FPI controller.
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Fig. 7, shows the MAE for the proposed PFPI, PFLS and FPI controller. The MAE values for the PFPI controller is decreased while it increases with the FPI controller. Therefore, the PFPI controller is able
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to reduce the influence of the uncertainties due to the change in the system parameters.
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Fig. 6. Response of the nonlinear system (Task 1 - case2).
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Fig. 7: MAE for the nonlinear system (Task 2 - case 2).
Task 2: In this simulation task, we study the response of the controlled system under the effect of
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the random noise. Gaussian noise with standard normal distribution and scaling ratio 0.015 is added to the measured output y(k ) . The performance of the proposed PFPI controller is better than the PFLS and
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the FPI controller where the PFPI controller can effectively overcome the random noise rather than the FPI controller and the PFLS as shown in Fig.8.
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Fig. 8: Response of the nonlinear system (Task 2).
Task 3: This task shows the robustness of the proposed PFPI controller when the measured output contains random noise and there is uncertainty in the system parameters. Gaussian noise with standard normal distribution is added at time 100 sec to the measured output y p (k ) where the scaling ratio for this noise is 0.015. The uncertainties in the system parameters are added at time 100 sec with
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values a1 0.13, a2 0.13, a3 0.13 and b 0.13 . Fig. 9, shows the response of the controlled system for all controllers. The performance of the proposed PFPI controller is better than the PFLS and
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the FPI controller. The MAE for the FPI controller is increased which mean that the system becomes unstable after adding the uncertainty values and random noise as shown in Fig. 10. In addition, the
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MAE for the proposed PFPI controller is lower than that obtained for the PFLS. So, the proposed PFPI controller is superior to reduce the effect of uncertainties in the presence of random noise on the
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measured output rather than the PFLS and the FPI controller.
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Fig. 9. Response of the nonlinear system (Task 3).
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Fig. 10: MAE for the nonlinear system (Task 3). Table 6, lists the performance comparison of the FPI controller, the PFLS, and the proposed PFPI
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controller. The obtained RMSE, MAE, and MAPE values for the proposed PFPI controller are lower than that obtained for the PFLS and the FPI controller. This demonstrates that the capability of the PFPI controller based on the PFS for controlling nonlinear systems with stochastic and non-stochastic
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uncertainties.
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Table 6: The RMSE, MAE and MAPE values for example 1.
Cases of
RMSE
Study
FPI PFLS [21]
MAE PFPI FPI PFLS [21]
PFPI 0.022
FPI PFLS [21] 6.0025
3.1954
PFPI
Task 1 (case 1) 0.174
0.1329
0.1232 0.060
Task 1 (case 2)
0.240
0.1407
0.1259 0.1677 0.0391 0.0230
16.7726 3.9055 2.2985
Task 2
0.1932
0.1391
0.1133 0.0687 0.0435 0.0283
6.5735
Task 3
0.3199
0.1915
0.1235 0.2059 0.0865 0.0371
20.4387 8.6471 3.7089
4.3543
2.1993
2.8328
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0.032
MAPE
Example 2: This example shows the applicability of the PFPI controller for engineering applications. The proposed PFPI controller is designed for controlling a continuous stirred tank reactor (CSTR) system. More details about this system can be found in [32]. The mathematical model for the CSTR
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system is described as:
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dx1 1 x1 (1 1 ) x1 ( 2 2 ) x22 dt dx2 2 2 x2 (1 1 ) x1 ( 2 2 ) x2 ( 3 3 )d 2 (t ) x2 u dt dx3 2 x3 ( 3 3 )d 2 (t ) x2 dt
y x3
(40)
(41)
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where 1 k1d1V / F , a2 k 2 d1V / F , 3 k3V / F , d1 is a constant activity and d 2 possesses timevarying behavior. The conversation of reactant A , the concentrations of middle reactant B , and the
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product C are denoted by x1 , x2 and x3 , respectively. With considering, the initial values for the parameters of the system are set as 1 (0) 3, 2 (0) 0.5, 3 (0) 1 . The control signal value is limited
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as u 0,1. The control objective is to make the output y x3 to track the desired output yd 0.5 without steady state error.
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The proposed PFPI controller performance is evaluated based on three simulation tasks including
the effect of uncertainties in the system’s parameters, time-varying nature of the controlled system, and the effect of the random noise. The simulation tasks for the proposed PFPI controller are compared with that of the PFLS and the FPI controller.
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Task 1: The uncertainty in the parameter 3 ( 3 0.15) is added at t 100 sec and the uncertainty is increased with value ( 3 0.25) at t 200 sec. The system response for this task is shown in Fig. 11. The response of the PFPI controller has a small overshoot and settling time compared with the PFLS and the FPI controller. So, the proposed PFPI controller can overcome the uncertainty
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due to the changing in the system parameters better than the PFLS and the FPI controller.
Fig. 11: Response of the CSTR process (Task 1).
Task 2: In this simulation task, we indicate the ability of the PFPI controller when the system parameters are varied with time. The process parameter d 2 (t ) is characterized by a time-varying behavior according to:
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d 2 (t ) 1 2 ( t 100) ) d 2 (t ) 1 0.5(1 e 2 ( t 220) ) d 2 (t ) 1.5 0.5(1 e
0 t 100 100 t 220
(41)
t 220
The performance of all controllers for this task is shown in Fig. 12. The response of the proposed PFPI controller has a good performance compared with the PFLS and the FPI controller. Therefore, the
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proposed PFPI controller is a robustness controller for time-varying systems.
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Fig. 12: Response of the CSTR process (Task 2).
Task 3: In this simulation task, we study the response of the CSTR system under the effect of the
random noise. Gaussian noise with standard normal distribution and scaling ratio 0.015 is added to the measured output y p (k ) . From Fig. 13, the performance of the proposed PFPI controller is better than the PFLS and the FPI controller where the PFPI controller can effectively overcome the random noise rather than other controllers. Fig. 14, shows that the MAE for the PFPI controller is lower than that of
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the PFLS and the FPI controller. So, this task indicates that the proposed PFPI controller is able to
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reduce the influence of noise.
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Fig. 13: Response of the CSTR process (Task 3).
Fig. 14: MAE of the CSTR process (Task 3).
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Table 7, lists the performance comparison of the FPI controller, the PFLS and the proposed PFPI controller in terms of RMSE, MAE, and MAPE. It is shown that the obtained values for RMSE, MAE and MAPE for the proposed PFPI controller are smaller than the PFLS and the FPI controller. Therefore, the proposed PFPI controller is able to reduce the effect of stochastic and non-stochastic uncertainties in the CSTR process.
Cases of Study
RMSE FPI
MAE
PFLS [21]
PFPI
FPI
PFLS [21]
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Table 7: RMSE, MAE, and MAPE for example 2.
MAPE
PFPI
FPI
PFLS [21]
PFPI
0.0452 0.0424
0.0399
0.0150 0.0110
0.0073
2.9979
2.1982
1.4571
Task 2
0.0464 0.0435
0.040
0.0159 0.0122
0.0077
3.1829
2.4378
1.5382
Task 3
0.0777 0.0635
0.049
0.0656 0.0319
0.0222
13.1273 6.3852 4.4443
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Task 1
5. Conclusion
In this paper, we introduced the PFPI controller, which combined the fuzzy logic system with
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probability theory. The proposed controller is designed for controlling uncertain nonlinear dynamic systems. The analytical structure, which finds the mathematical input/output relationship for the PFPI
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controller, has been derived. The results of the analytical structure prove that the proposed PFPI controller is a nonlinear PI controller. The gains for the proposed PFPI controller are a nonlinear function with the controller parameters. In addition, the proposed PFPI controller has six control
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parameters free to design, which means that the proposed controller has a more degree of freedom compared with the FPI controller. In this study, the sufficient conditions, which guarantee the system is
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BIBO stable are derived based on the small gain theorem. The proposed PFPI controller is applied to two uncertain nonlinear systems including the CSTR system. To indicate the robustness of the PFPI
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controller, the simulation results are compared with the PFLS and the FPI controller. Three simulation tasks have been applied to both controllers including the effect of the uncertainty in the system parameters, time-varying parameters and the effect of an external noise. The results indicate that the PFPI controller is better than other controllers. Three performance indices are measured in order to compare the proposed controller with other controllers. The measured errors for the proposed PFPI controller are lower than that obtained for other controllers. Finally, the proposed controller has advantages rather than other controllers, which are summarized as: 1) the proposed PFPI controller has more degree of freedom. 2) the performance of the proposed PFPI controller has lower RMSE, ISE and 22
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MAE. 3) the proposed PFPI controller is able to reduce the effect of stochastic and non-stochastic uncertainties in nonlinear dynamic systems. In the future work, we will use an adaption algorithm to update the parameters of the proposed controller to increase the robustness of the controller.
References
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[1] Y. Li, S. Tong, L. Liu, G. Feng, Adaptive output-feedback control design with prescribed performance for switched nonlinear systems, Automatica 80 (2017) 225-231.
[2] Y. Li, S. Tong, Y. Liu, T. Li, Adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on small-gain approach, IEEE Transactions on Fuzzy Systems 22
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(2014) 164-176.
[3] V. Kumar, A. P. Mittal, Parallel fuzzy P+ fuzzy I+ fuzzy D controller: design and performance evaluation, International Journal of Automation and Computing 7 (2010) 463-471. [4] F. Lin, R. Wai, S. Wang, A Fuzzy Neural Network Controller for Parallel-Resonant Ultrasonic
M
Motor Drive, IEEE Transactions on Industrial Electronics, 45 (1998) 928-937.
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[5] J. X. Xu, C. C. Hang, C. Liu, Parallel structure and tuning of a fuzzy PID controller, Automatica 36 (2000) 673-684.
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[6] F. Mrad, G. Deeb, Experimental comparative analysis of adaptive fuzzy logic controllers, IEEE Trans. on Control Systems Technology 10 (2002) 250–255.
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[7] J. Carvajal, G. Chen, H. Ogmen, Fuzzy PID controller: Design, performance evaluation, and
AC
stability analysis, Information Sciences 123 (2000) 249-270. [8] A. M. El-Nagar, M. El-Bardini, Hardware-in-the-loop simulation of interval type-2 fuzzy PD controller for uncertain nonlinear system using low cost microcontroller, Applied Mathematical Modelling 40 (3), 2346-2355. [9] E. Natsheh, K. A. Buragga, Comparison between conventional and fuzzy logic PID controllers for controlling DC motors, Int J Comput Sci. 7 (2010) 128–134. 23
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[10] A. Fadaei, K. Salahshoor, Design and implementation of a new fuzzy PID controller for networked control systems, ISA Trans. 47 (2008) 351–361. [11] T. P. Blanchett, G. C. Kember, R. Dubay, PID gain scheduling using fuzzy logic, ISA Trans. 39 (2000) 317–325.
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[12] A. M. El-Nagar, M. El-Bardini, Simplified interval type-2 fuzzy logic system based on new typereduction. Journal of Intelligent and Fuzzy Systems, 27 (2014) 1990-2010.
[13] M. El-Bardini, A. M. El-Nagar, Interval type-2 fuzzy PID controller: analytical structures and stability analysis. Arabian Journal for Science & Engineering 39 (2014) 7443–7458.
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[14] A. H. Meghdadi, M. R. Akbarzadeh-T, Probabilistic fuzzy logic and probabilistic fuzzy systems, in Proc. 10th IEEE Int. Conf., 3 (2001) 1127–1130.
[15] Z. Liu, H. X. Li, A probabilistic fuzzy logic system for modeling and control, IEEE Trans. Fuzzy Syst. 13 (2005) 848–859.
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[16] Y. Li, S. Sui, S. Tong, Adaptive fuzzy control design for stochastic nonlinear switched systems
ED
with arbitrary switchings and unmodeled dynamics, IEEE transactions on cybernetics 47 (2017) 403-414.
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[17] L. A. Zadeh, Discussion: Probability theory and fuzzy logic are complimentary rather than competitive, Technometrics 37 (1995) 271–276.
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[18] M. Laviolette, J. W. Seaman, Unity and diversity of fuzziness/spl minus/from a probability viewpoint, IEEE Transactions on Fuzzy Systems 2 (1994) 38-42.
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[19] H. X. Li, Z. Liu, A probabilistic neural-fuzzy learning system for stochastic modeling, IEEE Trans. Fuzzy Syst. 16 (2008) 898– 908.
[20] Z. Liu and H. X. Li, Probabilistic fuzzy logic system: A tool to process stochastic and imprecise information, in Proc. IEEE Int. Conf. Fuzzy Syst. (2009) 848–853.
24
ACCEPTED MANUSCRIPT
[21] W. J. Huang, G. Zhang and H. X. Li, A novel probabilistic fuzzy set for uncertainties-based integration inference. 2012 IEEE International Conference on Computational Intelligence for Measurement Systems and Applications (CIMSA), (2012) 58-62. [22] S. Hengjie, C. Miao, Z. Shen, W. Roel, D. H. Maja, C. Francky, A probabilistic fuzzy approach to
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modeling nonlinear systems, Neurocomputing 74 (2011) 1008–1025. [23] G. Zhang, H. X. Li, M. Gan, Design a wind speed prediction model using probabilistic fuzzy system, IEEE Transaction on Industrial Informatics 8 (2012) 819-827.
[24] W. J. Huang, Asymmetry-width probabilistic fuzzy logic system for rigid-flexible manipulator
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modeling, Journal of Chemical and Pharmaceutical Research 6 (2014) 753-758.
[25] C. Chen, T. Xiao, Probabilistic fuzzy control of mobile robots for range sensor based reactive navigation, Intelligent Control and Automation 2 (2011) 77-85.
[26] F. J. Lin, K. H. Tan, Squirrel-cage induction generator system using probabilistic fuzzy neural
M
network for wind power applications, IEEE International Conference on Fuzzy Systems (2015)1-8.
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[27] F. J. Lin, K. C. Lu, T. H. Ke, Probabilistic wavelet fuzzy neural network based reactive power control for grid-connected three-phase PV system during grid faults, Renewable Energy 92 (2016)
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437- 449.
[28] G. Zhang, H. X. Li, An efficient configuration for probabilistic fuzzy logic system, IEEE Trans.
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Fuzzy Syst. 20 (2012) 898-909. [29] H. Ying, Deriving analytical input–output relationship for fuzzy controllers using arbitrary input
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fuzzy sets and Zadeh Fuzzy AND operator, IEEE Trans. Fuzzy Syst. 14 (2006) 654–662.
[30] H. Ying, A general technique for deriving analytical structure of fuzzy controllers that use arbitrary trapezoidal/triangular input fuzzy sets and Zadeh fuzzy logic AND operator, Automatica 39 (2003) 1171–1184. [31] M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed., Prentice-Hall, Engle Wood, 1993. 25
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[32] W. Wu, Y. Chou, Adaptive feed forward and feed back control of nonlinear time-varying
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CE
PT
ED
M
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uncertain systems, International Journal of Control 72 (1999) 1127–1138.
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Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems Omar Shaheen , Ahmad M. El-Nagar , Mohammad El-Bardini , Nabila M. El-Rabaie PII: DOI: Reference:
S0016-0032(18)30023-1 10.1016/j.jfranklin.2017.12.015 FI 3267
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
22 March 2017 30 October 2017 4 December 2017
Please cite this article as: Omar Shaheen , Ahmad M. El-Nagar , Mohammad El-Bardini , Nabila M. El-Rabaie , Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems, Journal of the Franklin Institute (2018), doi: 10.1016/j.jfranklin.2017.12.015
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Probabilistic Fuzzy Logic Controller for Uncertain Nonlinear Systems
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Omar Shaheen1, Ahmad M. El-Nagar2, Mohammad El-Bardini3 and Nabila M. El-Rabaie4 Department of Industrial Electronics and Control Engineering Faculty of Electronic Engineering, Menoufia University Menouf, 32852, Egypt 1 [email protected] 2 [email protected] 3 [email protected] 4 [email protected]
Abstract:
This paper proposes a probabilistic fuzzy proportional - integral (PFPI) controller for controlling
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uncertain nonlinear systems. Firstly, the probabilistic fuzzy logic system (PFLS) improves the capability of the ordinary fuzzy logic system (FLS) to overcome various uncertainties in the controlled dynamical systems by integrating the probability method into the fuzzy logic system. Moreover, the input/output relationship for the proposed PFPI controller is derived. The resulting structure is equivalent to nonlinear PI controller and the equivalent gains for the proposed PFPI controller are a
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nonlinear function of input variables. These gains are changed as the input variables changed. The sufficient conditions for the proposed PFPI controller, which achieve the bounded-input bounded-
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output (BIBO) stability are obtained based on the small gain theorem. Finally, the obtained results indicate that the PFPI controller is able to reduce the effect of the system uncertainties compared with
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the fuzzy PI (FPI) controller. Keywords
Fuzzy logic controller, Probabilistic fuzzy logic system, stochastic uncertainties, Probabilistic fuzzy
1. Introduction
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controller, Analytical structure, BIBO stability
Most practical engineering systems are complex and time-variant with nonlinearity. These systems
contain a time-delay, an external disturbance and a modeling error. These problems reduce the performance of the systems [1, 2]. It is undesirable to be worthily controlled these systems using the classical controllers [3]. The design of modern controllers such as adaptive controllers contains complex mathematical analysis and also, have many difficulties in controlling time-varying and highly nonlinear plants [4]. For these reasons, many researchers have attempted to use intelligent controllers such as fuzzy control and neural network control in order to improve the system performance [4 - 7]. 1
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The presence of the system uncertainties increases the merits of the use of the fuzzy control as compared to other controllers due to the capability of fuzzy reasoning to reduce the effects of system uncertainties [8 - 13]. There are exist different uncertainties accompanied with the practical processes. These uncertainties may be stochastic and non-stochastic uncertainties [14]. The uncertainties due to the plant dynamics can be represented by non-stochastic uncertainties, but the uncertainties due to random noise
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in the measured data can be considered as stochastic uncertainties [15, 16]. Practically, most real world applications contain both stochastic and non-stochastic uncertainties and it is difficult to control such systems due to increasing uncertainties. However, the ordinary FLC (Type-1 fuzzy logic controller; T1FLC), which uses the ordinary fuzzy sets could not handle stochastic uncertainty and consequently, the T1FLC is not the effective controller for controlling the processes with stochastic and non-
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stochastic uncertainties [15].
The fuzzy logic system and the probability theory have been combined together in a valuable strategy to process uncertainties [17, 18]. The probabilistic fuzzy logic system (PFLS) is introduced for reducing the effect of stochastic and non-stochastic uncertainties. The PFLS uses the probabilistic
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fuzzy set (PFS) that is characterized via a three-dimensional membership function for capturing stochastic feature. The additional stochastic expression describes the secondary probability density
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function of the random membership grades [8, 19 - 21]. Therefore, the PFS provides more degrees of freedom that make the PFLS is an effective tool to reduce the effect of system uncertainties. Similar to FLSs, the PFLS contains three operations, which are a fuzzification process, an inference engine and a
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defuzzification operation [15]. However, in the PFLS, the reasoning mechanism is integrated with the probabilistic method. The PFLS was proposed and applied for system modeling [15, 19, 22 - 24],
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control of the process with stochastic uncertainties [15, 25 - 27] and function approximation [28]. Most of the researchers focused on the PFLS for system modeling. However, the research about PFLS for
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controlling the uncertain systems still exists. To date, there is no analytical structure and stability analysis for the PFLS to show the effect of the parameters of the probability density function on the system performance. The main purpose of this paper is developing the PFPI controller taking into its consideration the
advantages of the PFS to overcome the stochastic and non-stochastic uncertainties. This paper attempts to discuss the analytical structure of the proposed PFPI controller, which is an efficient method to show the input/output relationship for the proposed controller [29, 30]. Deriving the analytical structure of 2
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the probabilistic fuzzy logic controller (PFLC) is complicated than the T1FLC where the PFLC contains more degree of freedom and more complicated inference engine. The sufficient conditions, which achieve the BIBO stability for the proposed PFPI controller, are derived based on the small gain theorem. The proposed controller is applied for controlling two uncertain nonlinear systems. These are nonlinear mathematical plant and the continuous stirred tank reactor (CSTR) process. Simulation results indicate that the PFPI controller is a robust controller, which able to handle various uncertainties
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in the controlled system.
Our study has several contributions, which summarized as: (1) Probabilistic fuzzy PI controller using probabilistic fuzzy set is proposed to control uncertain nonlinear systems. (2) Deriving the analytical structure for the proposed PFPI controller to obtain the input/output relationship for the
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proposed controller. (3) Driving the stability conditions of the PFPI controller.
The rest of this paper is organized as follows: The structure of the PFPI controller is introduced in section 2. The analytical structure of the proposed PFPI controller is introduced in section 3. The stability analysis of the PFPI controller is presented in section 4. The simulation results are presented in section 5. Finally, the conclusions and the relevant references are presented.
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2. Probabilistic Fuzzy PI Controller
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The block diagram, which describes the main components of the proposed PFPI controller is illustrated in Fig. 1. It consists of three input/output variables. The two inputs are the error ( e(k ) ) and
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the change of error ( v(k ) ) and the output is the change of control signal ( U PFPI (k ) ).
e(k )
en (k )
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Ge
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Probabilistic Fuzzy Logic System
v(k )
Gv
vn (k )
Primary Fuzzification
Fuzzy Rules Traditional Defuzzification
Mamdani Inference
Probabilistic Processing
Bayesian Inferenc e
Probability Calculation Probabilistic Fuzzifcation
Probabilistic Defuzzifcation
Inference
Fig.1: Main components of the PFPI controller.
The two input variables have been scaled as follows: 3
U (k )
Gu
u (k )
uPFPI (k ) z 1
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en (k ) Ge (r (k ) ym (k ))
(1)
vn (k ) Gv (e(k ) e(k 1))
(2)
where r (k ) is the reference input, ym (k ) is the measured output, and k is the sampling instance. Ge and Gv are the scaling factors for the input variables e(k ) and v(k ) , respectively. The control signal
uPFPI (k ) is given as:
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uPFPI (k ) uPFPI (k 1) Gu U PFPI (k )
(3)
where uPFPI (k 1) is the previous value of uPFPI (k ) , U PFPI (k ) is the change in the denormalized output, and Gu is the output scaling factor.
The processing configuration of the PFPI controller consists of four major operations: the probabilistic fuzzification, rule base, the probabilistic fuzzy inference engine and the probabilistic
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defuzzification. Briefly, the proposed controller works as follows: In the probabilistic fuzzification process, the crisp input is fuzzified and the membership grade can be obtained by the primary fuzzification. The randomness uncertainties of the input data can be handled through probabilistic calculation. Then, the probabilistic fuzzy inference engine is used to infer the probabilistic fuzzy output
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based on the fuzzy rule-based. Finally, the crisp output can be obtained using the probabilistic defuzzification that consists of two parts; an ordinary defuzzification and probabilistic processing.
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2.1 Probabilistic fuzzification
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~ The PFS A can be expressed by the probability space ( x , , P) and it is represented in Fig. 2 [15]. ~ A ( x , , P) x X
(4)
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where x is the set of all possible events, is the field and P is the probability defined on . For a crisp input, the membership grade in the ordinary fuzzification is a single value. However, in
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the probabilistic fuzzification, it is a random variable with a certain probabilistic distribution function (PDF). The probabilistic fuzzification consists of two parts; the first part is the primary fuzzification, which obtains the membership grades ( x)0,1 of the crisp inputs in the antecedent part of the rules and the probabilistic fuzzification for calculating the corresponding probability of all possible membership grades for the controller inputs. In the probability calculation process, the Bayes' theorem processes the probabilistic information taking into account the group of fuzzy grades being independent variables. The probability function 4
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that is used for calculating the corresponding probability of all possible fuzzy membership grades is a Gaussian function. The probability can be calculated as: ( mi ) 2
Pi ( ) e
i2
(5)
where mi and i are the mean and standard deviation of the Gaussian function, respectively. To reduce the computation load, the fuzzy membership grades are divided into three cells i 1,2,3 and
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the weights mi and i are all set to constants. In this paper, i is set to be 1 [26, 27]. The joint probability of the membership grade is calculated as the following based on the Bayes' probability theorem: 3
P( ) Pi ( )
(6)
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i 1
P( )
(x) Primary MF
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1
Probabilistic density function
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P( )
(x)
(x) x
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Crisp input
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Fig.2: Probabilistic Fuzzy Set (PFS).
In this paper, the controller inputs en (k ) and vn (k ) are represented by two PFSs. The primary
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fuzzy sets are described by two triangle membership function called “positive” and “negative”, where the primary membership function of the controller inputs are described in Eqs. (7, 8) in which the constants q1 and q2 are design parameters.
e P n
(en (k ) q1 ) / 2q1 1 0
q1 en (k ) q1 en (k ) q1
e N n
otherwise
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1 (q1 en (k )) / 2q1 0
en (k ) q1 q1 en (k ) q1 other wis e
(7)
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v P n
(vn (k ) q2 ) / 2q2 1 0
q2 vn (k ) q2
VN
vn (k ) q2 otherwise
1 (q2 vn (k )) / 2q2 0
vn (k ) q2 q2 vn (k ) q2
(8)
otherwise
2.2 Rule-based The universe of discourse for the controller output U PFPI (k ) is divided into three overlapping
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PFSs where the primary fuzzy set is represented by triangular membership function labeled N (Negative), Z (Zero) and P (Positive). The rule-based for the proposed PFPI controller is described in Table 1.
Table 1: Rule-based for the proposed PFPI controller.
Derivative of error signal P N
Error Signal
N
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2.3 Probabilistic fuzzy inference
Z N
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The inference engine produces fuzzy output based on the rule-based. The probabilistic frameworks are used to perform the probabilistic fuzzy inference. The AND operations in the fuzzy rule-based are
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represented by the minimum operation (min ()). Thus, the firing grades, which are the result of the minimum operator for the four fuzzy rules are obtained as:
R~1 min( e P , v P ) for U PI (k ) P
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n
n
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R~ 2 min( e P , v N ) for U PI (k ) Z n
n
(9) (10)
R~3 min( e N , v P ) for U PI (k ) Z
(11)
R~ 4 min( e N , v N ) for U PI (k ) N
(12)
n
n
n
n
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The probability of the firing grade R~j can be obtained as in Eq. (13) based on the Bayes’
inference method:
P( R~j ) P(~en j ) P(v~n j )
(13)
where P( R~j ) , P( e~n j ) and P( v~n j ) denote the corresponding probability of the firing grade R~j , ~en j and v~n j ; with considering ~en j and v~n j as independent events. 6
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2.4 Probabilistic defuzzification This operation is used to obtain the crisp output based on the probabilistic processing method. It uses the PFSs instead of the ordinary fuzzy sets in the output of the proposed PFPI controller. The output domain of the PFS is divided into N points and the output of the centroid defuzzification can be obtained as:
y
l 1 N
l
R ( x, y l )
l 1
R
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N
y
( x, y l )
(14)
where y and R are random variables. To obtain the crisp output for the proposed PFPI controller, we perform the mathematical expectation on Eq. (14) as:
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U PFPI (k ) EX ( y)
(15)
One of the defuzzification approaches is concerned with calculation the mathematical expectation and the crisp output is obtained as:
N
l 1 N
M
U PFPI (k )
C E ( l
E ( l 1
R
R
)
)
C N E ( N ) CZ E ( Z ) C P E ( P ) E ( N ) E (Z ) E (P )
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U PFPI (k )
(16)
(17)
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where C N , CZ and C P are the centroid for three output fuzzy sets. The mathematical expectation of
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the fuzzy output variables can be obtained as: E (P ) R~1 P(R~1 )
E ( Z ) R~ 2 P( R~ 2 ) R~3 P( R~3 ) E (N ) R~ 4 P(R~ 4 )
(18) (19) (20)
Assume CZ 0 , then: U PI (k )
C N E ( N ) CP E ( P ) E( N ) E(Z ) E( P )
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(21)
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3. Analytical Structure of the Probabilistic Fuzzy PI Controller In this section, we present the analytical structure for the proposed PFPI controller. The min() operators in the four fuzzy rule-based, which defined in Eqs. (9-12) are performed to obtain the mathematical input/output relationship of the proposed PFPI controller. Assume that en (k ) is inside
[q1 , q1 ] and vn (k ) is inside [q2 , q2 ] for brevity. The final values for the four fuzzy rules in Eqs. (9-12)
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and their corresponding probabilities are first calculated to derive the proposed controller output, which defined in Eq. (21). After that, we can calculate the mathematical expectation of the fuzzy output variables from Eqs. (18–20) and finally, the crisp output of the proposed PFPI controller can be obtained. To obtain which membership value is smaller in each fuzzy rule, the input space
[q1 , q1 ] [q2 , q2 ] is divided into two regions in each rule, each region is called input combination
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(IC). However, all the four rules are executed at the same time. We obtain a total 4 ICs by superimposing all rules as shown in Fig. 3. Table 2, indicates the result of the firing grades and their corresponding probabilities for these 4 ICs.
vn P
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vn N
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q2
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vn (k )
2
3
1 4
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q2 en N
en P
q1
q1
Fig. 3. Input space division.
8
en (k )
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Table 2: The firing grades and their corresponding probabilities of the four rules for IC1 to IC4.
Rule 1
R~1
R~ 3
P( R~ 3 )
R~ 4
P( R~ 4 )
v N
e g2
e N
e g3
e N
e g4
e P
e g1
v N
e g2
e N
e g3
v N
e g4
e P
e g1
e P
e g2
v P
e g3
v N
e g4
v P
e g1
e P
e g2
v P
e g3
e N
e g4
n
n
4
P( R~ 2 )
e g1
n
3
R~ 2
Rule 4
v P
1 2
P( R~1 )
Rule 3
n
n
n
n
n
where the parameters g1, g2 , g3 and g 4 are defined as:
n
n
n
n
n
n
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IC No.
Rule 2
n
n
g1 3 Pen (2m1 2m2 2m3 ) Pen 3 Pvn (2m1 2m2 2m3 ) Pvn (2m12 2m22 2m32 ) 2
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2
g 2 3 Pen (2m1 2m2 2m3 ) Pen 3 Nvn (2m1 2m2 2m3 ) Nvn (2m12 2m22 2m32 ) 2
2
(22) (23)
g 3 3 Nen (2m1 2m2 2m3 ) Nen 3 Pvn (2m1 2m2 2m3 ) Pvn (2m12 2m22 2m32 )
(24)
g 4 3 Nen (2m1 2m2 2m3 ) Nen 3 Nvn (2m1 2m2 2m3 ) Nvn (2m12 2m22 2m32 )
(25)
2
2
2
2
For each IC, the mathematical expectation is calculated from Eqs. (18-20) and then, the outputs of
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these equations are used in Eq. (21) to obtain the output of the controller. After some mathematical operations, the mathematical expressions of the proposed PFPI controller are obtained for each IC.
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Finally, the mathematical expressions of u PI (k ) for all the ICs that shown in Fig. 3 share the same
C1r Ge e(k ) C2r Gv v(k ) C3r D1r Ge e(k ) D2r Gv v(k ) D3r
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structure pattern as: r u PFPI (k ) Gu
r 1,...,4
(26)
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where Cn and Dn , for n 1,2,3 are constants. The values of these constant depend on the parameters of the proposed PFPI controller.
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Letting
K pr (e(k ), v(k ))
Gu C2r Gv D1r Ge e(k ) D2r Gv v(k ) D3r
K ir (e(k ), v(k ))
Gu C1r Ge D1r Ge e(k ) D2r Gv v(k ) D3r
Gu C3r (e(k ), v(k )) r D1 Ge e(k ) D2r Gv v(k ) D3r r
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Then r uPFPI (k ) K pr (e(k ), v(k )) v(k ) Kir (e(k ), v(k )) e(k ) r (e(k ), v(k ))
(27)
In order to simplify the analysis, the parameters of the proposed PFPI are assumed as:
q1 q2 q , CP R, CN R where R 0 . Tables 3 and 4 indicate the values of the proportional and integral gains for the proposed PFPI controller in each IC, respectively. The proportional and integral
uFPPI (k ) ( K p Ki ) e(k ) K p e(k 1)
Letting
K1P (e(k ), v(k )) ( K p Ki ) , and K 2P (e(k ), v(k )) K p
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gains are a nonlinear function of controller parameters. Equation (27) can be rewritten in general as:
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uPFPI (k ) K1P (e(k ), v(k )) e(k ) K 2P (e(k ), v(k )) e(k 1) (e(k ), v(k ))
(28)
(29)
Equation (30) is defined the digital form of the linear PI controller as:
uPId (k ) uPId (k ) uPId (k 1) K1d e(k ) K 2d e(k 1)
(30)
where K1d and K 2d are the digital linear PI controller parameters. From Eq. (29) and Eq. (30), we find that the proposed PFPI controller is a nonlinear PI controller in an incremental form with the variable
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gains K1P (e(k ), v(k )) , K 2P (e(k ), v(k )) and the variable control offset (e(k ), v(k )) . Remark 1: The PFPI controller has six control parameters free to design ( Ge , Gv , GU , m1 , m2 , m3 ),
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whereas the FPI controller only has three parameters ( Ge , Gv , GU ). If these parameters are used in the designing of the proposed PFPI controller, we have more degree of freedom that makes the proposed
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controller are able to improve the system performance and reduce the effect of system uncertainties. Table 3: The dynamic proportional gain for the PFPI controller.
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IC No.
Kp
1
R Gv Gu e g1 (e g3 e g4 )Ge e (e g1 e g2 )Gv v q(e g1 e g2 e g3 e g4 )
2
R Gv Gu e g4 (e g1 e g3 )Ge e (e g2 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
3
R Gv Gu e g4 (e g1 e g2 )Ge e (e g3 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
4
RGv Gu e g1 (e g2 e g4 )Ge e (e g1 e g3 )Gv v q(e g1 e g2 e g3 e g4 )
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Table 4: The dynamic integral gain gains for the PFPI controller.
IC No.
Ki
2
R Ge Gu e g1 (e g1 e g3 )Ge e (e g2 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
3
R Ge Gu e g1 (e g1 e g2 )Ge e (e g3 e g4 )Gv v q(e g1 e g2 e g3 e g4 )
4
R Ge Gu e g4 (e g2 e g4 )Ge e (e g1 e g3 )Gv v q(e g1 e g2 e g3 e g4 )
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1
R Ge Gu e g4 (e g3 e g4 )Ge e (e g1 e g2 )Gv v q(e g1 e g2 e g3 e g4 )
4. Stability Analysis
The small gain theorem is used in this section to drive the sufficient conditions for the proposed PFPI controller to achieve the BIBO stability [31]. Fig. 4, shows the feedback system where the proposed PFPI controller is represented by subsystem S1 while the subsystem S 2 represents the
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controlled system. The following equations describe the overall feedback system:
u1
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e1 u1 y2 , e2 u2 y1 , y1 S1 e1 and y2 S 2 e2 . e1
+
y1
S1
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-
e2
y2
S2
+ +
u2
Fig. 4: Feedback control system.
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Assume the gain of S1 is 1 (S1 ) , and the gain of S 2 is 2 (S 2 ) . Assume that there are constants
1, 2 , 1 0 and 2 0 so that: y1 S1e1 1 e1 1
(31)
y2 S 2e2 2 e2 2
(32)
Based on the small gain theorem, the sufficient condition, which makes the controlled system is BIBO stable is 1 2 1 . A bounded output pair ( y1, y2 ) is produced at any bounded input pair (u1 , u2 ) . 11
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Let we assume the controlled system is nonlinear described by N . The feedback system that shown in Fig. 4 is obtained by defining r (k ) u1 (k ) , e(k ) e1 (k ) , uPFPI (k ) y1 (k ) , uPI (k 1) u2 (k ) ,
uPFPI (k ) e2 (k ) , y(k ) y2 (k ) in Fig. 1. The sufficient conditions for the proposed PFPI controller, which achieve the BIBO stability for each IC is derived as the following: According to Eq. (29), we obtain the following equation as:
where M e is the maximum magnitude of e(k ) that described as: M e : sup e(k ) k 0
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uPFPI (k ) y1 (k ) H1e1 (k ) K1 e1 (k ) K 2 M e
(33)
(34)
According to Eq. (31) and Eq. (33), we have 1 K1 . Then, the following equation is obtained as:
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y(k ) y2 (k ) H 2e2 (k ) Ne2 (k ) N e2 (k )
(35)
By comparing Eq. (35) with Eq. (32), we have 2 N . Therefore, the controlled system based on the proposed PFPI controller is BIBO stable if the parameters of the proposed controller must satisfy
defined as:
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the inequality K1 N 1 . Where K1 is defined for the regions IC1 to IC4 as in Table 5, where M v is
M v : sup v(k ) sup v(k ) v(k 1)
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k 0
Table 5: Values of
K1
RGu ( Ge e g4 Gv e g1 ) (e g3 e g4 )Ge M e (e g1 e g2 )Gv M v q(e g1 e g2 e g3 e g4 )
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1
K1 for stability conditions of the PFPI controller.
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IC No.
(36)
k 0
2
RGu ( Ge e g1 Gv e g4 ) (e g1 e g3 )Ge M e (e g2 e g4 )Gv M v q(e g1 e g2 e g3 e g4 )
3
RGu ( Ge e g1 Gv e g4 ) (e g1 e g2 )Ge M e (e g3 e g4 )Gv M v q(e g1 e g2 e g3 e g4 )
4
RGu ( Ge e g4 Gv e g1 ) (e g2 e g4 )Ge M e (e g1 e g3 )Gv M v q(e g1 e g2 e g3 e g4 )
The sufficient condition for the proposed PFPI controller, which achieve the BIBO stability is obtained by combining together all the conditions that shown in Table 5. 12
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Remark 2: The sufficient condition of the BIBO stability for the proposed PFPI controller is dependent on the system parameters ( Ge , Gv , GU , m1 , m2 , m3 ). Therefore, the values of these parameters must be chosen to make the system gains achieve the condition of the BIBO stability.
4. Simulation Results To show the robustness of the PFPI controller, we present two nonlinear systems. The simulation
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results of the proposed PFPI controller are compared with that of the FPI controller and PFLS, which is proposed previously in [21]. The comparison between two controllers is based on the same number of the membership functions, the number of rules and the same scaling factors. The control system performance is compared in terms of the root-mean-squared errors (RMSE), the mean absolute errors (MAE) and the mean absolute percentage errors (MAPE), which are given as:
MAE
1 N
y (k ) y (k ) k 1
k 1
(37)
N
1 N N
2
d
y (k ) y (k )
(38)
yd (k ) y(k ) 100 % yd (k )
(39)
k 1
d
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MAPE
N
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1 N
RMSE
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Example 1: This example demonstrates the performance of the proposed PFPI controller for a nonlinear system. Consider the nonlinear system is defined by the following:
(a1 a1 ) y p (k 1) y p (k 2) y p (k 1) (a2 a2 )
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y p (k )
(a3 a3 ) y p (k 1) 2 y p (k 2) 2
(b b)u(k 1)
(40)
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where the normal values of the system parameters are a1 1, a2 1.5, a3 1and b 1 . a1 , a2 , a3 and
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b are the uncertainty values due to the uncertainty in the parameters of the system. y p (k ) and u (k )
are the nonlinear system output and the control signal, respectively. Simulation tasks are performed with sampling time 0.1 sec. Several simulation tasks are performed to show the robustness of the proposed PFPI controller. The set-point is a unit step input for all the simulation tasks. Task 1: This simulation task indicates the performance of the PFPI controller under the effect of the uncertainty in system’s parameters. There are two cases as the following:
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Case
1:
The
uncertainties
in
the
system’s
parameters
with
the
values:
(a1 0.13, a2 0.13, a3 0.13, b 0.13) are added at time equal 100 sec. The response of the
controlled system for the proposed PFPI, PFLS and FPI controllers is shown in Fig.5. The proposed PFPI controller can provide a robust performance after adding the uncertainty values. The performance of the PFPI controller has smaller overshoot and settling time than that obtained for PFLS and FPI
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controller.
Fig. 5. Response of the nonlinear system (Task 1 - case 1). 2:
The
uncertainties
in
the
system’s
parameters
with
the
values:
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Case
(a1 0.14, a2 0.14, a3 0.14, b 0.14) are added at time equal 100 sec. The system response
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for this case is shown in Fig.6. it’ clear that the controlled system remains stable after adding the uncertainty values with the proposed PFPI controller but it becomes unstable with the FPI controller.
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Fig. 7, shows the MAE for the proposed PFPI, PFLS and FPI controller. The MAE values for the PFPI controller is decreased while it increases with the FPI controller. Therefore, the PFPI controller is able
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to reduce the influence of the uncertainties due to the change in the system parameters.
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Fig. 6. Response of the nonlinear system (Task 1 - case2).
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Fig. 7: MAE for the nonlinear system (Task 2 - case 2).
Task 2: In this simulation task, we study the response of the controlled system under the effect of
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the random noise. Gaussian noise with standard normal distribution and scaling ratio 0.015 is added to the measured output y(k ) . The performance of the proposed PFPI controller is better than the PFLS and
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the FPI controller where the PFPI controller can effectively overcome the random noise rather than the FPI controller and the PFLS as shown in Fig.8.
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Fig. 8: Response of the nonlinear system (Task 2).
Task 3: This task shows the robustness of the proposed PFPI controller when the measured output contains random noise and there is uncertainty in the system parameters. Gaussian noise with standard normal distribution is added at time 100 sec to the measured output y p (k ) where the scaling ratio for this noise is 0.015. The uncertainties in the system parameters are added at time 100 sec with
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values a1 0.13, a2 0.13, a3 0.13 and b 0.13 . Fig. 9, shows the response of the controlled system for all controllers. The performance of the proposed PFPI controller is better than the PFLS and
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the FPI controller. The MAE for the FPI controller is increased which mean that the system becomes unstable after adding the uncertainty values and random noise as shown in Fig. 10. In addition, the
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MAE for the proposed PFPI controller is lower than that obtained for the PFLS. So, the proposed PFPI controller is superior to reduce the effect of uncertainties in the presence of random noise on the
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measured output rather than the PFLS and the FPI controller.
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Fig. 9. Response of the nonlinear system (Task 3).
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Fig. 10: MAE for the nonlinear system (Task 3). Table 6, lists the performance comparison of the FPI controller, the PFLS, and the proposed PFPI
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controller. The obtained RMSE, MAE, and MAPE values for the proposed PFPI controller are lower than that obtained for the PFLS and the FPI controller. This demonstrates that the capability of the PFPI controller based on the PFS for controlling nonlinear systems with stochastic and non-stochastic
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uncertainties.
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Table 6: The RMSE, MAE and MAPE values for example 1.
Cases of
RMSE
Study
FPI PFLS [21]
MAE PFPI FPI PFLS [21]
PFPI 0.022
FPI PFLS [21] 6.0025
3.1954
PFPI
Task 1 (case 1) 0.174
0.1329
0.1232 0.060
Task 1 (case 2)
0.240
0.1407
0.1259 0.1677 0.0391 0.0230
16.7726 3.9055 2.2985
Task 2
0.1932
0.1391
0.1133 0.0687 0.0435 0.0283
6.5735
Task 3
0.3199
0.1915
0.1235 0.2059 0.0865 0.0371
20.4387 8.6471 3.7089
4.3543
2.1993
2.8328
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0.032
MAPE
Example 2: This example shows the applicability of the PFPI controller for engineering applications. The proposed PFPI controller is designed for controlling a continuous stirred tank reactor (CSTR) system. More details about this system can be found in [32]. The mathematical model for the CSTR
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system is described as:
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dx1 1 x1 (1 1 ) x1 ( 2 2 ) x22 dt dx2 2 2 x2 (1 1 ) x1 ( 2 2 ) x2 ( 3 3 )d 2 (t ) x2 u dt dx3 2 x3 ( 3 3 )d 2 (t ) x2 dt
y x3
(40)
(41)
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where 1 k1d1V / F , a2 k 2 d1V / F , 3 k3V / F , d1 is a constant activity and d 2 possesses timevarying behavior. The conversation of reactant A , the concentrations of middle reactant B , and the
PT
product C are denoted by x1 , x2 and x3 , respectively. With considering, the initial values for the parameters of the system are set as 1 (0) 3, 2 (0) 0.5, 3 (0) 1 . The control signal value is limited
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as u 0,1. The control objective is to make the output y x3 to track the desired output yd 0.5 without steady state error.
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The proposed PFPI controller performance is evaluated based on three simulation tasks including
the effect of uncertainties in the system’s parameters, time-varying nature of the controlled system, and the effect of the random noise. The simulation tasks for the proposed PFPI controller are compared with that of the PFLS and the FPI controller.
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Task 1: The uncertainty in the parameter 3 ( 3 0.15) is added at t 100 sec and the uncertainty is increased with value ( 3 0.25) at t 200 sec. The system response for this task is shown in Fig. 11. The response of the PFPI controller has a small overshoot and settling time compared with the PFLS and the FPI controller. So, the proposed PFPI controller can overcome the uncertainty
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due to the changing in the system parameters better than the PFLS and the FPI controller.
Fig. 11: Response of the CSTR process (Task 1).
Task 2: In this simulation task, we indicate the ability of the PFPI controller when the system parameters are varied with time. The process parameter d 2 (t ) is characterized by a time-varying behavior according to:
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d 2 (t ) 1 2 ( t 100) ) d 2 (t ) 1 0.5(1 e 2 ( t 220) ) d 2 (t ) 1.5 0.5(1 e
0 t 100 100 t 220
(41)
t 220
The performance of all controllers for this task is shown in Fig. 12. The response of the proposed PFPI controller has a good performance compared with the PFLS and the FPI controller. Therefore, the
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proposed PFPI controller is a robustness controller for time-varying systems.
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Fig. 12: Response of the CSTR process (Task 2).
Task 3: In this simulation task, we study the response of the CSTR system under the effect of the
random noise. Gaussian noise with standard normal distribution and scaling ratio 0.015 is added to the measured output y p (k ) . From Fig. 13, the performance of the proposed PFPI controller is better than the PFLS and the FPI controller where the PFPI controller can effectively overcome the random noise rather than other controllers. Fig. 14, shows that the MAE for the PFPI controller is lower than that of
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the PFLS and the FPI controller. So, this task indicates that the proposed PFPI controller is able to
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reduce the influence of noise.
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Fig. 13: Response of the CSTR process (Task 3).
Fig. 14: MAE of the CSTR process (Task 3).
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Table 7, lists the performance comparison of the FPI controller, the PFLS and the proposed PFPI controller in terms of RMSE, MAE, and MAPE. It is shown that the obtained values for RMSE, MAE and MAPE for the proposed PFPI controller are smaller than the PFLS and the FPI controller. Therefore, the proposed PFPI controller is able to reduce the effect of stochastic and non-stochastic uncertainties in the CSTR process.
Cases of Study
RMSE FPI
MAE
PFLS [21]
PFPI
FPI
PFLS [21]
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Table 7: RMSE, MAE, and MAPE for example 2.
MAPE
PFPI
FPI
PFLS [21]
PFPI
0.0452 0.0424
0.0399
0.0150 0.0110
0.0073
2.9979
2.1982
1.4571
Task 2
0.0464 0.0435
0.040
0.0159 0.0122
0.0077
3.1829
2.4378
1.5382
Task 3
0.0777 0.0635
0.049
0.0656 0.0319
0.0222
13.1273 6.3852 4.4443
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Task 1
5. Conclusion
In this paper, we introduced the PFPI controller, which combined the fuzzy logic system with
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probability theory. The proposed controller is designed for controlling uncertain nonlinear dynamic systems. The analytical structure, which finds the mathematical input/output relationship for the PFPI
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controller, has been derived. The results of the analytical structure prove that the proposed PFPI controller is a nonlinear PI controller. The gains for the proposed PFPI controller are a nonlinear function with the controller parameters. In addition, the proposed PFPI controller has six control
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parameters free to design, which means that the proposed controller has a more degree of freedom compared with the FPI controller. In this study, the sufficient conditions, which guarantee the system is
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BIBO stable are derived based on the small gain theorem. The proposed PFPI controller is applied to two uncertain nonlinear systems including the CSTR system. To indicate the robustness of the PFPI
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controller, the simulation results are compared with the PFLS and the FPI controller. Three simulation tasks have been applied to both controllers including the effect of the uncertainty in the system parameters, time-varying parameters and the effect of an external noise. The results indicate that the PFPI controller is better than other controllers. Three performance indices are measured in order to compare the proposed controller with other controllers. The measured errors for the proposed PFPI controller are lower than that obtained for other controllers. Finally, the proposed controller has advantages rather than other controllers, which are summarized as: 1) the proposed PFPI controller has more degree of freedom. 2) the performance of the proposed PFPI controller has lower RMSE, ISE and 22
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MAE. 3) the proposed PFPI controller is able to reduce the effect of stochastic and non-stochastic uncertainties in nonlinear dynamic systems. In the future work, we will use an adaption algorithm to update the parameters of the proposed controller to increase the robustness of the controller.
References
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[1] Y. Li, S. Tong, L. Liu, G. Feng, Adaptive output-feedback control design with prescribed performance for switched nonlinear systems, Automatica 80 (2017) 225-231.
[2] Y. Li, S. Tong, Y. Liu, T. Li, Adaptive fuzzy robust output feedback control of nonlinear systems with unknown dead zones based on small-gain approach, IEEE Transactions on Fuzzy Systems 22
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(2014) 164-176.
[3] V. Kumar, A. P. Mittal, Parallel fuzzy P+ fuzzy I+ fuzzy D controller: design and performance evaluation, International Journal of Automation and Computing 7 (2010) 463-471. [4] F. Lin, R. Wai, S. Wang, A Fuzzy Neural Network Controller for Parallel-Resonant Ultrasonic
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Motor Drive, IEEE Transactions on Industrial Electronics, 45 (1998) 928-937.
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[5] J. X. Xu, C. C. Hang, C. Liu, Parallel structure and tuning of a fuzzy PID controller, Automatica 36 (2000) 673-684.
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[6] F. Mrad, G. Deeb, Experimental comparative analysis of adaptive fuzzy logic controllers, IEEE Trans. on Control Systems Technology 10 (2002) 250–255.
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[7] J. Carvajal, G. Chen, H. Ogmen, Fuzzy PID controller: Design, performance evaluation, and
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stability analysis, Information Sciences 123 (2000) 249-270. [8] A. M. El-Nagar, M. El-Bardini, Hardware-in-the-loop simulation of interval type-2 fuzzy PD controller for uncertain nonlinear system using low cost microcontroller, Applied Mathematical Modelling 40 (3), 2346-2355. [9] E. Natsheh, K. A. Buragga, Comparison between conventional and fuzzy logic PID controllers for controlling DC motors, Int J Comput Sci. 7 (2010) 128–134. 23
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[10] A. Fadaei, K. Salahshoor, Design and implementation of a new fuzzy PID controller for networked control systems, ISA Trans. 47 (2008) 351–361. [11] T. P. Blanchett, G. C. Kember, R. Dubay, PID gain scheduling using fuzzy logic, ISA Trans. 39 (2000) 317–325.
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[12] A. M. El-Nagar, M. El-Bardini, Simplified interval type-2 fuzzy logic system based on new typereduction. Journal of Intelligent and Fuzzy Systems, 27 (2014) 1990-2010.
[13] M. El-Bardini, A. M. El-Nagar, Interval type-2 fuzzy PID controller: analytical structures and stability analysis. Arabian Journal for Science & Engineering 39 (2014) 7443–7458.
AN US
[14] A. H. Meghdadi, M. R. Akbarzadeh-T, Probabilistic fuzzy logic and probabilistic fuzzy systems, in Proc. 10th IEEE Int. Conf., 3 (2001) 1127–1130.
[15] Z. Liu, H. X. Li, A probabilistic fuzzy logic system for modeling and control, IEEE Trans. Fuzzy Syst. 13 (2005) 848–859.
M
[16] Y. Li, S. Sui, S. Tong, Adaptive fuzzy control design for stochastic nonlinear switched systems
ED
with arbitrary switchings and unmodeled dynamics, IEEE transactions on cybernetics 47 (2017) 403-414.
PT
[17] L. A. Zadeh, Discussion: Probability theory and fuzzy logic are complimentary rather than competitive, Technometrics 37 (1995) 271–276.
CE
[18] M. Laviolette, J. W. Seaman, Unity and diversity of fuzziness/spl minus/from a probability viewpoint, IEEE Transactions on Fuzzy Systems 2 (1994) 38-42.
AC
[19] H. X. Li, Z. Liu, A probabilistic neural-fuzzy learning system for stochastic modeling, IEEE Trans. Fuzzy Syst. 16 (2008) 898– 908.
[20] Z. Liu and H. X. Li, Probabilistic fuzzy logic system: A tool to process stochastic and imprecise information, in Proc. IEEE Int. Conf. Fuzzy Syst. (2009) 848–853.
24
ACCEPTED MANUSCRIPT
[21] W. J. Huang, G. Zhang and H. X. Li, A novel probabilistic fuzzy set for uncertainties-based integration inference. 2012 IEEE International Conference on Computational Intelligence for Measurement Systems and Applications (CIMSA), (2012) 58-62. [22] S. Hengjie, C. Miao, Z. Shen, W. Roel, D. H. Maja, C. Francky, A probabilistic fuzzy approach to
CR IP T
modeling nonlinear systems, Neurocomputing 74 (2011) 1008–1025. [23] G. Zhang, H. X. Li, M. Gan, Design a wind speed prediction model using probabilistic fuzzy system, IEEE Transaction on Industrial Informatics 8 (2012) 819-827.
[24] W. J. Huang, Asymmetry-width probabilistic fuzzy logic system for rigid-flexible manipulator
AN US
modeling, Journal of Chemical and Pharmaceutical Research 6 (2014) 753-758.
[25] C. Chen, T. Xiao, Probabilistic fuzzy control of mobile robots for range sensor based reactive navigation, Intelligent Control and Automation 2 (2011) 77-85.
[26] F. J. Lin, K. H. Tan, Squirrel-cage induction generator system using probabilistic fuzzy neural
M
network for wind power applications, IEEE International Conference on Fuzzy Systems (2015)1-8.
ED
[27] F. J. Lin, K. C. Lu, T. H. Ke, Probabilistic wavelet fuzzy neural network based reactive power control for grid-connected three-phase PV system during grid faults, Renewable Energy 92 (2016)
PT
437- 449.
[28] G. Zhang, H. X. Li, An efficient configuration for probabilistic fuzzy logic system, IEEE Trans.
CE
Fuzzy Syst. 20 (2012) 898-909. [29] H. Ying, Deriving analytical input–output relationship for fuzzy controllers using arbitrary input
AC
fuzzy sets and Zadeh Fuzzy AND operator, IEEE Trans. Fuzzy Syst. 14 (2006) 654–662.
[30] H. Ying, A general technique for deriving analytical structure of fuzzy controllers that use arbitrary trapezoidal/triangular input fuzzy sets and Zadeh fuzzy logic AND operator, Automatica 39 (2003) 1171–1184. [31] M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed., Prentice-Hall, Engle Wood, 1993. 25
ACCEPTED MANUSCRIPT
[32] W. Wu, Y. Chou, Adaptive feed forward and feed back control of nonlinear time-varying
AC
CE
PT
ED
M
AN US
CR IP T
uncertain systems, International Journal of Control 72 (1999) 1127–1138.
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