A servo system control with time-varying and ... - Semantic Scholar
Applied Soft Computing 11 (2011) 5735–5744
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A servo system control with time-varying and nonlinear load conditions using type-2 TSK fuzzy neural system Erdal Kayacan a,∗ , Yesim Oniz a , Ayse Cisel Aras a , Okyay Kaynak a , Rahib Abiyev b b
Bogazici University, Department of Electric and Electronics Engineering, 34342, Bebek, Istanbul, Turkey Near East University, Department of Computer Engineering, Mersin, Turkey
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 November 2010 Received in revised form 7 March 2011 Accepted 11 March 2011 Available online 4 April 2011
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A type-2 Takagi-Sugeno-Kang fuzzy neural system is proposed and its parameter update rules are derived using fuzzy clustering and gradient learning algorithms. The proposed type-2 fuzzy neural system is used for the control and the identification of a real-time servo system. Fuzzy c-means clustering algorithm is used to determine the initial places of the membership functions to ensure that the gradient descent algorithm used afterwards converges in a shorter time. A number of different load conditions including nonlinear and time-varying ones are used to investigate the performance of the proposed control algorithm. The control structure has the ability to regulate the servo system with reduced oscillations when compared with the results of its type-1 counterpart around the set point signal in the presence of load disturbances. © 2011 Elsevier B.V. All rights reserved.
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Keywords: Type-2 fuzzy neural system Gradient-based learning algorithm Fuzzy c-means clustering Servo system
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1. Introduction
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DC motors are often used in industrial control applications where speed control is required over a wide range. Moreover, they can deliver three or more times their rated torque momentarily, and supply over five times rated torque in emergency situations. One of the important features of DC motors is that the speed of the motor can be controlled smoothly down to zero. Moreover, they can respond quickly to changes in control signals due to their high ratio of torque to inertia [1]. If the parameters of a DC motor can be obtained precisely, then its control would be a relatively straightforward problem and model-based approaches (PID, pole placement, etc.) could be used. However, in much of the real world the information we have is, or is allowed to be, imprecise, uncertain, incomplete, unreliable, partially true or partially possible [2]. In general, noise in the measurements and the changes in the parameters due to the current and temperature variations give rise to uncertainties. Furthermore, the load characteristics of the DC motor are often nonlinear, such as in centrifugal fans, pumps and blowers. In such cases, not only does the performance of the model-based approaches drastically decrease, but also the complexity of the controller design increases.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (E. Kayacan), [email protected] (Y. Oniz), [email protected] (A.C. Aras), [email protected] (O. Kaynak), [email protected] (R. Abiyev). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.03.008
There are many situations in industrial control systems when control engineers face the difficulty of incomplete or insufficient information. In such cases, model-free approaches are preferred. The most common approaches to model-free design in the literature are artificial neural networks (ANNs) and fuzzy logic systems (FLSs). When the learning capability of ANNs is combined with the uncertainty-handling ability of the FLSs, the resulting structure is called fuzzy neural networks (FNNs). The application of FNNs for constructing a DC motor speed controller allows an increase in the performance of the control system [3,4]. The structures of the FLSs mentioned above can be based on TSK or Mamdani-type fuzzy reasoning mechanisms. The membership functions used in the antecedent and the consequent parts (in the case of Mamdani types) are generally type-1 that are designed and trained from numeric data sets or linguistic information. However, when there are uncertainties associated with information in the knowledge base of the process, then type-2 fuzzy sets can be utilized to handle the effects of the uncertainties in the rule base. In this study, a type-2 FNS structure is developed for such a purpose and its efficacy is verified by experimental studies on the setup DR300 AMIRA [5], by means of which time-varying loads can be generated with any non-linearity at will. One of the most important problems in the design of type-2 FNS is the learning of the system parameters. In this paper, the learning rules are derived for the adaptation of the controller parameters using fuzzy clustering and gradient algorithms. The knowledge base of the designed structures is based on type-2 TSK fuzzy rules, the antecedent parts of which are characterized by type-2 fuzzy sets, and the consequent parts use crisp linear functions. The paper is organized as
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E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
follows: In the following section, a literature review is given. In Section 3, the mathematical description of the DC servo system is presented. In Section 4, type-2 fuzzy sets and the inference mechanism of the type-2 fuzzy TSK system are briefly introduced. In Section 5, the parameter update rules based on the clustering and the gradient algorithms are derived. In Section 6, the simulation and the experimental studies are presented for both the control and the identification of the DC motor system in the presence of load disturbances. Finally, in Section 7, conclusions are given.
Fig. 1. A servo system setup.
2. Literature review
Fig. 2. Block diagram of the motor with load.
Description
Armature terminal voltage Induced electromotive force Armature current Armature winding resistance Armature winding inductance Motor constant Magnetic excitation Speed of the rotor Load torque Acceleration torque Torque produced by the motor Moment of inertia of the motor Electrical time constant Mechanical time constant
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UA E IA RA LA C ω ML MB M J TA TM
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Name
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Table 1 Nomenclature.
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The combination of fuzzy systems and neural networks has recently become a popular approach in engineering fields to solve control, identification, prediction, pattern recognition and etc. One well known structure is the adaptive neuro-fuzzy inference system (ANFIS). Others include TSK-type recurrent fuzzy networks and fuzzy wavelet neural networks [6] that have been widely used for identification and control purposes. In [7], a variable structure system theory based training procedure is proposed and in [8] its use in a neuro-adaptive scheme for the control of an anti-lock braking systems is described. A fuzzy sliding mode controller with a compensator is presented as a robust control scheme in [9]. A compensator based on the sliding mode control theory is used to improve the dynamical characteristics of the drive system. Type-2 fuzzy sets have been introduced by Zadeh as the extension of type-1 fuzzy sets. Mendel and Karnik have developed the theory of type-2 fuzzy sets further in [10]. The theoretical background of interval type-2 fuzzy system and its design principles are described in [11]. Some applications of the type-2 FNSs can be seen in the literature. A recent work [12] describes the design of a type-2 FNN. The optimal training algorithm derived in that paper is later corrected in [13]. In [14] the design of a type-2 FNN structure for modeling nonlinear systems is presented. This is further developed in [15] for nonlinear system identification and nonlinear time varying channel equalization. In [13–15], type-2 Gaussian membership functions with uncertain means and fixed standard deviations are used in the antecedent part. The consequent parts of these systems include weighting interval set with unity membership grade, i.e. an interval type-1 fuzzy set. The use of interval type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements, environmental noise and other external disturbances is discussed in [16]. In [17] the learning problems of type-2 fuzzy system and its use for controlling Hot Strip Temperature are considered. In [18] the design principles and learning of type-2 fuzzy systems are described. The type-2 FNN models used in [19,20] use type-2 fuzzy set with uncertain mean in antecedent part and type-1 interval TSK fuzzy sets in consequent parts. In [21] interval type-2 fuzzy sets are used to construct a type2 support vector machines fusion fuzzy logic system. In [22] the stability type-2 fuzzy control system is analyzed. Lin et al. [23] uses type-2 fuzzy set with uncertain mean in antecedent part and type1 interval fuzzy sets in consequent parts. Uncu and Turksen [24] proposes discrete interval valued type-2 fuzzy system models generated by a learning parameter of fuzzy c-means clustering. Hwang and Rhee [25] proposes type-2 fuzzy approach to clustering. In this paper, we use a type-2 fuzzy neuro system for DC motor control. Hybrid algorithm, which uses clustering and gradient techniques, is applied to learn the parameters of the fuzzy neuro system. Experiments have been done to demonstrate that the type-2 fuzzy neuro system can achieve robust characteristics and regulation performance with time-varying and nonlinear load conditions.
3. Mathematical description of the permanently excited DC motor The experimental setup used in this study [5] consists of two DC motors, which are connected by a mechanical clutch. The first motor is used for the control of the rotation speed or the shaft angle. The second one acts as a generator, by means of which nonlinear load conditions can be created (see Fig. 1). The block diagram of the servo system is shown in Fig. 2, the nomenclature of the symbols used being given in Table 1. From the block diagram of the system depicted in Fig. 2, the transfer function of the overall system can readily be derived as follows: ω(s) =
1 RA 1 1 + TA s UA (s) − ML (s) C 1 + TM s + TM TA s2 KM C 1 + TM s + TM TA s2
(1)
where TM =
JRA KM C
and
TA =
LA RA
The numerical values used in this study are: armature terminal voltage= 24 V; rated torque= 0.096 Nm; moment of inertia of the system= 80.45 × 10−6 kg m2 ; armature inductance= 3 mH; armature resistance= 3.13 ; back emf constant= 0.06 Vs; and torque constant= 0.06 Nm/A. 4. Type-2 fuzzy neural system structure Type-2 fuzzy systems are characterized by fuzzy IF-THEN rules. The parameters in the antecedent and the consequent parts of the rules include type-2 fuzzy values. In Gaussian type-2 fuzzy sets
E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
a
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b
μ(x)
μ(x) μ(x) μ(x) σ1
σ
2
c1
c2
Fig. 3. Type-2 fuzzy set with uncertain STD (a) and uncertain mean (b).
(2)
˜ and x is à ˜ and . . . and x is à ˜ IF x1 is à m 2 j1 j2 jm
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where c and are the center and the width of the membership function, and x is the input vector. In this study membership functions with uncertain mean c ∈ [c1 , c2 ] are considered. The kernel of a type-2 fuzzy inference system is the fuzzy knowledge base. In a fuzzy knowledge base, the information that consists of input–output data points of the system is interpreted into linguistically interpretable fuzzy rules in IF-THEN form. In a type-2 fuzzy rule, both sides, i.e. the antecedent and the consequent parts, can be type-2 or just the antecedent part can be type-2. Mendel and his co-authors classify type-2 TSK fuzzy rules into three, depending on type-1 and type-2 nature of the antecedent membership functions and the parameters of the consequent part. In Model I, the membership functions used in the antecedent part are type-2, whereas the parameters of the consequent part are type-1 fuzzy sets. In many research studies the consequent part of (3) is taken as a type-1 fuzzy set [10,14,18–20,23]. In Model II, the membership functions used in the antecedent part are type-2, and the parameters of the consequent part are crisp numbers. In Model III, the membership functions used in the antecedent part are type-1, and the parameters of the consequent part are type-1 fuzzy sets (it is to be noted that this is not the conventional Mamdani type system as the weighted average of the inference output of each rule is taken as the output of the FLS, and this is a type-1 set, not a crisp number). This paper focuses on Model II.
m
1 (x − c)2 2 2
co
−
n.
(x) ˜ = exp
When the number of rules in a type-1 and a type-2 FNS are the same, then the number of design parameters in the latter system is more than the one in the former. The larger number of parameters increases the approximation capability of the type-2 FNS. When system data has a high level of noise and the behavior of the system is characterized by uncertainties, then a type-2 fuzzy system allows to model such systems with more accuracy than a type-1 fuzzy system. An increase in the computational load (as compared to type-1) may not be the case as a type-2 FNS may be able to describe the process using less number of fuzzy rules than a type-1 fuzzy system. In order to reduce the complexity, most researchers prefer to use interval type-2 fuzzy sets in designing type-2 fuzzy systems. Since the secondary memberships of the interval type-2 fuzzy systems are equal to one, the computations associated with interval type2 fuzzy sets are relatively manageable [26]. This is the approach followed in this paper too. Additionally, in order to further reduce the computational complexity, type-2 fuzzy sets are used in the antecedent parts of the rules and linear functions in the consequent part of the rules. The use of such a structure significantly simplifies the type reduction process as compared to the ordinary type-2 fuzzy systems. The rules used in this paper are such that the consequent parts are TSK type as indicated below:
ca
uncertainties can be associated with the mean and the standard deviation (STD). In Fig. 3(a) and (b) Gaussian type-2 fuzzy sets with uncertain STD and uncertain mean are shown. The mathematical expression for the membership function is expressed as:
THEN yj =
m
wij xi + bj
i=1
Fig. 4. Structure of type-2 TSK fuzzy neural system.
(3)
E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
i
i
Ãi (xk ) = [ i (xk ), ¯ Ãi (xk )] = [ , ¯ ] Ã
k
(4)
k
u=
(5)
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In the first layer of Fig. 4, the input signals are fed to the system. In the second layer each node corresponds to one linguistic term. In this layer, for each input signal entering the system, the membership degrees and ¯ are determined using (2). The third layer calculates the firing strengths of the rules which are realized using the product t-norm operator. f = (x1 ) × (x2 ) × . . . × (xn ) Ã1 Ã2 Ãn f¯ = ¯ Ã (x1 ) × ¯ Ã (x2 ) × . . . × ¯ Ãn (xn )
w
The fourth layer determines the outputs of the linear functions yi (i = 1, . . ., n), in the consequent parts.
yj =
m
wij xi + bj
(6)
i=1
The fifth, the sixth and the seventh layers perform the type reduction and the defuzzification operations. After determining the firing strengths of rules, the defuzzified output of the type-2 TSK fuzzy system is determined. The inference engine of type-2 TSK FNS is proposed in [27,28]. The final output of type-2 FNS is determined as:
u(x) = [uL (x), uR (x)] =
···
f1 ∈ [f ,f1 ] 1
1
f1 ∈ [f ,f1 ]
f j yj
j=1 N
N i=1
fi (x)yi /
N
fi (x)
(7)
i=1
1
where uL (x) and uR (x) can be computed using iterative Karnik–Mendel algorithm [26]. However the application of Karnik–Mendel algorithm may not be convenient for the control structure of a real time implementation. For this reason [28]
q +
fj
N
f¯j yj
j=1 N
j=1
(8) f¯j
j=1
where p and q are the design parameters that weight the sharing of lower and upper firing levels of each fired rule, N is the number of active rules. f¯j and f j are determined using (5), and yj is deter-
co
m
mined using (6). These parameters can be tuned during the design of the TSK system. For example, [29] uses the least mean square method for finding the values of p and q parameters. In Fig. 4, layer 5 computes the product of the membership degrees f and f¯ and linear functions yj . Layer 6 includes two summation blocks. One of these blocks computes the sum of the output signals from the layer 5 (nominator part of (8)) and the other block computes the sum of the output signal of the layer 4 (denominator part of (8)). Layer 7 calculates output of the network using (8). After the calculation of the output signal in the type-2 FNS, the training of the network is started. The training includes an adjustment of the parameters c1ij , c2ij and ij in the membership functions in the second layer and the parameters wij , bj (i = 1, . . ., m, j = 1, . . ., n) of the linear functions in the fourth layer. In the next section, the parameter update rules of type-2 FNS are derived. 5. Parameter update rules The design of type-2 FNS includes the determination of the unknown parameters in the antecedent and the consequent parts of the fuzzy IF-THEN rules. The parameter update rules are derived for a type-2 TSK FNS with fixed mean and uncertain STD. In this paper fuzzy clustering technique is applied for the determination of the parameters in the antecedent part for modeling purposes and the gradient algorithm is used for the learning of the parameters in the antecedent and the consequent part for modeling and control purposes. In the antecedent part, the input space is divided into a set of fuzzy regions, and in the consequent part the system behavior in those regions is described. Recently a number of different approaches have been used for designing fuzzy IF-THEN rules based on clustering [25,30–32], table look-up scheme [33,34], least-squares method (LSM) [14,27], recursive least-squares (RLS) method [17], gradient algorithms [7,9,1,29,12–14,18–20,35], and genetic algorithms [6,35,36]. In this paper, the fuzzy clustering is applied to design the antecedent parts, and the gradient algorithm is applied to design the consequent parts of the fuzzy rules. The aim of clustering methods is to identify a certain group of data from a large data set, such that a concise representation of the behavior of the system is produced. Each cluster center can be translated into a fuzzy rule for identifying the class. Clustering has been used for type-1 fuzzy systems [31,32]. For type-2 fuzzy systems, subtractive clustering and fuzzy clustering have been developed [32]. Subtractive clustering is an unsupervised clustering method, in which the number of clusters for input data points is determined by the clustering algorithm. Sometimes the number of clusters in the input space should be determined in advance. In these cases, the supervised clustering algorithms are of primary concern. Fuzzy c-means clustering is one of them. It can efficiently be used for type-1 fuzzy systems [31] owing to its simple structure and sufficient accuracy. However there are still uncertainties in the determination of the model structure, such as the number of rules and the number of variables involved in the rule premises. When
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k
N
p
ya
of the i th input defined as a Gaussian membership function. The parameters wij and bj (i = 1, . . ., m, j = 1, . . ., n) are the parameters in the consequent part of the rules. The structure of the multi input–single output type-2 FNS used in this study is given in Fig. 4. In this structure the input signals to the network are the external input signals x = {x1 , . . ., xn }. The kernel of the fuzzy inference system is the fuzzy knowledge base which consists of the input–output data points interpreted into fuzzy rules. The type-2 FNS is constructed using type-2 TSK fuzzy rules which are given by (3). The development of the type-2 FNS includes the determination of the proper values of unknown coefficients in the antecedent and the consequent parts of each rule. Let us consider the design of a type-2 FNS when the input membership functions are of Gaussian and given by (2). If both c and parameters of the Gaussian function are taken to be uncertain (within certain intervals), the parameter space of the system can become very large. In this paper, only one of these parameters is assumed to be uncertain, i.e. uncertain STD and fixed mean or fixed STD and uncertain mean. It is to be noted that the fixed values are also subject to parameter adjustment. Due to the uncertainties in the antecedent part, the output of the type-2 fuzzy rules will have uncertainties. Let us first assume that the parameters of membership functions are represented by an uncertain mean and a fixed STD. Each membership function of the antecedent part is represented by an upper and a lower mem¯ bership function. These are denoted with (x) ¯ and (x), or A(x) and A(x).
proposes an alternative approach. In this paper, we use inference engine for type-2 TSK system proposed in [28], which is given as:
n.
where x1 , x2 , . . ., xm are input variables, yj (j = 1, . . ., n) are output ˜˜ denotes a type-2 membership function for j th rule variables, A ij
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E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
rd
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and Jm2 =
C N
2 d2 um ij ji
(10)
i=1 j=1
2 1 d (ui − ui ) 2
∂E ∂¯cij
(13)
(14)
(15)
n.
∂u ∂f¯j ∂j ∂u ∂f j ∂ij + ∂f j ∂ ∂ij ¯ ij ∂ij ∂f¯j ∂ ij
j
j
¯ ij ∂u ∂f¯j ∂ ∂u ∂f j ∂ij + ¯j ∂ ∂f j ∂ ∂c1ij ¯ ij ∂c1ij ∂ f ij ¯ ij ∂u ∂f j ∂ij ∂u ∂f¯j ∂ + ¯ ∂f j ∂ ∂c2ij ∂ ¯ ∂c2 ∂fj ij ij ij
(16)
(17)
The parameters of the type-2 FNS can thus be updated using (12)–(14) together with (15)–(17). As mentioned above, the parameter p and q in (8) enable us to adjust the lower or the upper portions in the final output. In this paper the values of p and q are optimized during learning from an initial value of 0.5 using: p(t + 1) = p(t) −
∂E ∂p
(18)
q(t + 1) = q(t) −
∂E ∂q
(19)
where
where 1 ≤ m1 ≤ m2 ≤ ∞. Fuzzy partitioning is carried out through an iterative optimization of the objective functions (10), with the update of membership function and the cluster centers. For a detailed analysis, the reader can refer to [25]. After the design of the antecedents parts by fuzzy clustering, the gradient descent algorithm is applied to design the consequent parts of the fuzzy rules. In what follows, the parameter update rules are derived for a type-2 TSK FNS. At the first step, the output error is calculated: O
E=
j
∂E ∂E = ∂c2ij ∂u
w
i=1 j=1
w
Jm1 =
1 d2 , um ij ji
∂E ∂E = ∂c1ij ∂u
lower and the upper membership of the data point. Updating of the cluster centers are performed by the extension of interval type-2 fuzzy sets while executing fuzzy c-means algorithm. The interval type-2 fuzzy set is incorporated into fuzzy c-means clustering. The use of the fuzzifiers m1 and m2 result in two different objective functions to be minimized. C N
∂E ∂E ∂u ∂yj = ∂wij ∂u ∂yj ∂wij ∂E ∂u ∂yj ∂E = ∂bj ∂u ∂yj ∂bj
ca
(9)
where djk (dik ) denotes the distance between cluster j(k) and data point xi , C is the number of clusters and (xi ) and ¯ j (xi ) are the j
(12)
where is the learning rate, m is the number of input signals of the network (input neurons) and n is the number of rules (hidden neurons). The derivatives in (12) are determined by the following formulas:
∂E ∂E = ∂ij ∂u
k=1
k=1
∂E ∂bj
c2ij (t + 1) = c2ij (t) −
∂E ∂ij
co
1 ⎪ , otherwise ⎪ ⎪ C ⎪ ⎪ ⎪ 2/(m −1) 2 ⎪ (dik /djk ) ⎪ ⎩ k=1 ⎧ 1 1 1 ⎪ , if < ⎪ ⎪ C C C ⎪ ⎪ ⎪ 2/(m1 −1) ⎪ (dik /djk ) (dik /djk ) ⎪ ⎨ 1 ⎪ , otherwise ⎪ ⎪ C ⎪ ⎪ ⎪ 2/(m −1) 2 ⎪ (dik /djk ) ⎪ ⎩
ij (t + 1) = ij (t) −
∂E ; ∂c ij
bj (t + 1) = bj (t) −
The derivatives in (13) and (14) are determined by the following formulas:
k=1
k=1
∂E ; ∂wij
c1ij (t + 1) = c1ij (t) −
ya
¯ j (xi ) =
wij (t + 1) = wij (t) −
ka
j
k=1
al
(xi ) =
⎧ 1 1 1 ⎪ , if ≥ ⎪ ⎪ C C C ⎪ ⎪ ⎪ 2/(m1 −1) ⎪ (dik /djk ) (dik /djk ) ⎪ ⎨
The parameters wij , bj and c1ij , c2ij and ij (i = 1, . . ., m, j = 1, . . ., n) are adjusted using the following formulas:
m
fuzzy clustering is used in type-2 fuzzy systems, the computational complexity is reduced and a better distribution of the cluster centers is obtained. In this paper, the clustering technique proposed in [25] is used for structuring the premise part of the fuzzy system. It is to be noted that fuzzy c-means does not guarantee classification for a data set that contains clusters with different densities, that have different volumes with different numbers of data points. Because of the imperfect information about the data points, there exists uncertainties in the various parameters that are used for the assignment of fuzzy memberships. In [25], the representation and the management of these uncertainties are handled by varying the fuzzifier (parameter m) in the membership function. The fuzzifier m controls the amount of fuzziness in fuzzy classification. Consequently, the input data set is extended into interval type-2 fuzzy sets. For an input data point xi , the highest and the lowest memberships are defined using different fuzzy degrees m1 and m2 and the footprint of uncertainty is thus created. The primary memberships that extend data point xi by interval type-2 fuzzy sets are determined as [25]
5739
(11)
i=1
where O is number of output signals of the network (in the given case O = 1), udi and ui are the desired and the current output values of the network, respectively.
fj ∂E = (u − ud ) n ∂p
(20) fj
j=1
f¯j ∂E = (u − ud ) n ∂q
(21) f¯j
j=1
where ¯ yj − u ∂u (yj − u) ∂u ∂E =p n ; =q n = u(t) − ud (t); ∂u ∂f j ∂f¯j f¯j fj j=1
j=1
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E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
; u¯ =
f¯j yj
j=1 n
fj
j=1
(22) f¯j
j=1
If t-norm prod operator is used, then ∂f j
N1
=
∂
ij
∂f¯j
; kj
∂ ¯ ij
k=1,k = / i
N1
=
¯ kj
(23)
k=1,k = / i
where i = 1, . . ., N1, k = 1, . . ., N1, and j = 1, . . ., N2. For t-norm min operator: if ( − ) ≥ 0, then kj
ij
∂f
j
∂
= 1, else
ij
¯ kj − ¯ ij ) ≥ 0, if (
then
∂f
j
∂
= 0,
ij
∂f¯j ∂ ¯ ij
= 1, else
∂f¯j ∂ ¯ ij
= 0;
where i = 1, . . ., N1, k = 1, . . ., N1, and j = 1, . . ., N2. Upper and lower membership functions between i th input and j th hidden neurons of layer 3 can be written as follows:
⎧ ⎨ G(c2 , , x ), x ≤ c1ij + c2ij ij ij i i 2 (x) = ij ⎩ G(c1 , , x ), x > c1ij + c2ij ij ij i i 2
ca
¯ ij (x) =
(24)
xi < c1ij c1ij ≤ xi ≤ c2ij xi > c2ij
ya
G(c1ij , ij , xi ), 1, G(c2ij , ij , xi ),
where G(cij , ij , xi ) is determined as: G(cij , ij , xi ) = exp
1 (xi − cij ) − 2 2
2
(25)
j
∂c1ij
∂ ¯ j (xi ) ∂c2ij ∂ (xi ) j
∂c2ij
∂ ¯ j (xi ) ∂ij
∂ (xi ) j
∂ij
=
=
=
=
=
⎪ ⎩ G(c1ij , ij , xi ) ⎧ 0, ⎪ ⎨ 0,
(xi − c1ij )
rd
ij
.e
⎪ ⎩ 0, ⎧ 0, ⎪ ⎨ 0,
c1ij ≤ xi ≤ c2ij xi > c2ij c1ij + c2ij xi ≤ 2 c1ij + c2ij xi > 2
w
∂ (xi )
=
ij2
,
xi < c1ij c1ij ≤ xi ≤ c2ij
(xi − c2ij ) ⎪ , xi > c2ij ⎩ G(c2ij , ij , xi ) ij2 ⎧ (x − c2 ) c1 + c2ij ⎪ ⎨ G(c2ij , ij , xi ) i 2 ij , xi ≤ ij
⎪ ⎩ 0,
(26)
w
∂c1ij
w
¯ j (xi ) ∂
⎧ (xi − c1ij ) ⎪ , xi < c1ij ⎨ G(c1ij , ij , xi ) 2
al
ij
Then
One important problem in the learning algorithms is the convergence. The convergence of the gradient descent method depends on the selection of the initial value of the learning rate. Usually, this value is selected within the interval [0, 1]. A large value of the learning rate may lead to unstable learning, while a small value of the learning rate results in a slow learning speed. In this study, an adaptive approach is used to update this parameters. That is, the learning of the type-2 FNS parameters is started with a small value of the learning rate . During learning, is increased if the value of the change of error E = E(k) − E(k + 1) is positive, and decreased if negative. This strategy ensures a stable learning for the type-2 FNS, guarantees the convergence and speeds up the learning.
ka
Define c cluster centers arbitrarily for the Xi data set for all Xi data set do Compute membership function using (9) for m = m1 and m = m2 Centre update. Compute cl minimum (left) and cr maximum (right) of centre using iterative algorithm given in [24] Perform defuzification using c = (cl + cr )/2 if cold = cthen Save c cluster centers and stop clustering end if end for Spread cluster’s centers as c1 = c − c c2 = c + c Assign cluster centers to the centers of membership functions for all membership functions do Determine width of membership functions as (i, j) = max(abs(c2 (i, j) − c2 (i, j − 1)), abs(c2 (i, j) − c2 (i, j + 1))) (where i = 1, . . . , m; j = 1, . . . , n, m is number of input signals, n is number of rules) end for Randomly generate w and b parameters of consequent part of type-2 FNS: Set maximum epoch number max ep no, learning tag for antecedent part learn antecedant to 0 or 1, and epoch number ep no = 1; Set initial values of p = 0.5, q = 0.5 while ep no ≤ max ep nodo for all input data pairs do For given input data calculate UT 2FNS output of type-2 FNS using (2-8) Calculate error as e(t) = UT 2FNS − U d ; (where U d is the desired output) if abs(e(t)) > then Update w, b, p, q parameters using (12-23). (where is an acceptable small value) end if if learn antecedant = 1then Update c1 , c2 and parameters of membership functions using (24-28) end if end for end while Save parameters of type-2 FNS
m
n
The design steps of type-2 FNS for the identification problem are given below:
co
f j yj
j=1
u=
n
n.
n
ij
xi >
(27)
2 c1ij + c2ij
⎧ (x − c1 )2 ⎪ ⎪ G(c1ij , ij , xi ) i 3 i , xi < c1ij ⎪ ⎪ ij ⎨
2
c1ij ≤ xi ≤ c2ij
0,
⎪ ⎪ (xi − c2ij )2 ⎪ ⎪ , xi > c2ij ⎩ G(c2ij , ij , xi ) ij3 ⎧ (x − c2 )2 c1 + c2ij ⎪ ⎪ G(c2ij , ij , xi ) i 3 ij , xi ≤ ij ⎨ ij
2
(x − c1 ) c1 + c2ij ⎪ ⎪ ⎩ G(c1ij , ij , xi ) i 3 ij , xi > ij ij
2
2
(28)
6. Simulation and experimental studies In real life, the systems can only be modeled with some level of certainty. The uncertainties cannot be adequately described by deterministic models and therefore conventional control approaches based on such models are unlikely to result in the required performance. Under such conditions the use of soft computing methodologies can be a valuable alternative. Most industrial plants have nonlinear characteristics and are susceptible to internal and external disturbances. Furthermore, the time-varying nature of the plants may be interpreted as the uncertainties in the plant coefficients. This type of uncertainty could be described using fuzzy sets. In this study the FNS structure described in the previous section is used as the identifier and the adaptive controller for the control of the laboratory setup.
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5741
Fig. 5. The block diagram of the identifier.
6.1. Neuro-fuzzy system for the identification of the dynamical plants
0.5
co
u(k)
0.4
n.
0.35
ca
0.3
0.25
0
ka
al
0.45 pu
(29)
K
i
400
600
800
1000
Samples Fig. 6. The input of the identifier.
0.3
RMSE
0.25 0.2 0.15 0.1 0.05
The inputs to the type-2 FNS based identifier are the input signal u(k), and 1-step delayed values of the real plant outputs y(k). The problem is to find the proper parameter values for the type-2 FNS structure such that the difference between the plant output y(k) and the identifier output yn (k) will be minimized for all input values u(k). The training of the type-2 FNS is carried out for 200 epochs with 1000 time steps in each epoch. As a performance criterion the rootmean-square-error (RMSE) defined in (30) is used.
K 1 2 RMSE = (xd − xi )
200
0.35
rd
.e
w
w
0.6 pu
⎪ ⎪ ⎩ 0.55 pu
if 0 < k ≤ 150 if 150 < k ≤ 300 if 300 < k ≤ 450 if 450 < k ≤ 600 if 600 < k ≤ 750
w
Reference(k) =
⎧ 0.4 pu ⎪ ⎪ ⎨ 0.5 pu
m
0.45
ya
The dynamical process of finding the input–output relations for a system is called identification in the literature. In identification, the clustering involves the determination of clusters in data space and the translation of these clusters into fuzzy rules such that the model obtained is close to the identified system. For this purpose, first the fuzzy clustering approach is applied to the input data points of the plant in order to determine the cluster centers. The obtained cluster centers are then used in order to organize the premise parts of the fuzzy rules. It is to be noted that it would be possible not to use a clustering algorithm and spread the membership functions onto the universe of discourse randomly. However, in this case, the gradient descent algorithm would require more time to find the final places of the membership functions. In this study, the fuzzy c-means clustering algorithm is used to determine the initial places of the membership functions so that the gradient descent algorithm converges in a shorter time. In order to prevent the system from a possible saturation, the input–output data set is collected in a closed loop fashion (Fig. 5) using a conventional PI controller. The coefficients of the PI controller are KP = 1 and KI = 5. The sampling period of the real-time system is equal to 0.01 s. The period of the reference signal is 750 samples, and the mathematical expression for the reference signal can be seen in (29):
(30)
i=1
where K represents the total number of the samples. Fig. 6 shows the input to the identifier. While Fig. 7 shows the RMSE values versus epoch number, which indicates a stable learning with the gradient based algorithm, Fig. 8 demonstrates the
0 0
50
100
150
200
Epoch number Fig. 7. RMSE versus epoch number.
output of the model and the real-time system. As can be seen from Fig. 8, the type-2 FNS gives accurate modeling results. Identification is performed with 4 fuzzy rules. These fuzzy rules are constructed using two clusters that are determined for each input variables. Different combinations of these clusters are considered for the organization of the premise parts of the fuzzy rules. As a result of clustering of premise parts and training of consequent parts, the four fuzzy rules are generated and the parameters of the type-2 FNS are determined. After the determination of the parameters in the antecedent part, the gradient algorithm is applied for the learning of the parameters in the antecedent and the consequent part. These are the
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0.7
Tacho Speed (Normalized Bipolar)
0.7 target model output
0.6
Motor Speed
0.5 0.4 0.3 0.2 0.1 0
0.6 0.5 0.4 0.3 0.2 0.1 T1FNS T2FNS Ref
0 0
−0.1 0
200
400
600
800
5
10
15
20
Time (s)
1000
Samples
Fig. 10. The speed responses of the motor for T1FNS and T2FNS.
Fig. 8. Output of the model and the real-time system.
Table 3 Square of the error values at each time step of the control algorithm. Load type 2
Load type 3
1.330 0.956
1.021 0.953
0.856 0.841
ka
Load type 1
al
Type-1 FNS Type-2 FNS
−0.1 0 0.1
−0.5 0.5 1.5
e(k)
e(k)
−0.2 0.1 0.4
−0.05 0.05 0.15
e(k)
oij
e(k)
e(k)
0.2 0.2 0.2
0.1 0.1 0.1
w
−0.3 0 0.3
e(k)
−0.3 0.7 1.7
w
e(k)
c2ij
e(k)
0.3 0.3 0.3
w
e(k)
.e
rd
Table 4 Initial values of the centers and standard deviation of the input membership functions for the control case. c1ij
m
0.3620 0.5393 0.3688 0.5498
n.
0.2811 0.4074 0.2920 0.4079
The structure of the control scheme is shown in Fig. 9. The variable y(k) is the output signal of the plant, g(k) is the set-point signal, e(k), e(k) and e(k) are the error, the change in the error and the sum of the error, respectively. D indicates differentiation operation and indicates integration operation. AMIRA DR300 DC motor experimental setup is used to test and compare the performances of the type-1 FNS and type-2 FNS controllers. The sampling time is set to 20 ms for all the experiments. The speed of the motor and the load torque are scaled to [−1, 1]. In order to determine the efficiency and the accuracy of the proposed controllers, three different types of load conditions are considered. Fig. 10 represents the speed responses of the motor for type-1 FNS and type-2 FNS controllers. The corresponding load condition starts with a value of 0.2 pu, and increases suddenly to 0.5 pu on 7th s, and then comes back to 0.2 pu on 14th s. As can be seen from Fig. 10, type-1 FNS and type-2 FNS adapt their parameters when the load on the motor changes suddenly in 7th s and in 14th s. In order to initialize wij , bj , c1ij , c2ij and ij , first the simulation model of the setup is run for 200 epochs. Fig. 11 shows the speed response of the motor under the sinusoidal load condition, i.e. ML (t) = 0.35 + 0.15sin (0.75 t). Besides, Fig. 12 shows the speed response of motor under load condition which is proportional to the square of the speed, i.e. ML (t) = 1.5(Speed)2 . This type of load corresponds to the load-torque characteristics of centrifugal fans, pumps and blowers. It can be inferred from Table 3 that type-2 FNS controller gives better performance than the type-1 FNS controller in all load conditions. In this table, load type 1, load type 2, and load type 3 correspond to the step change, sinusoidal, and proportional to the square of the speed of load torque, respectively. Tables 4 and 5 show the centers and standard deviations before and after control process, respectively.
ca
c2ij
Input 2
ya
c1ij
Input 1
6.2. Type-2 fuzzy neuro system for the control of a dynamical system
co
Table 2 The values of the centers of the input membership functions for the identification case after clustering.
parameters of the linear functions of the layer 4 in Fig. 4. The initial values of w, a and b are selected randomly within the interval [−1, 1]. Table 2 shows the centers of the type-2 membership functions after the clustering process. The number of rows shows the number of membership functions for each input.
Fig. 9. Structure of type-2 TSK fuzzy neural system.
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Table 5 Final values of the centers and standard deviation of the input membership functions for the control case. c1ij
c2ij −0.1006 0.0596 0.2550
−0.5004 0.6010 1.5050
oij
−0.2631 −0.1253 0.3543
−0.0896 −0.0215 0.1442
0.1022 −0.0959 0.1740
0.0735 0.1276 −0.0418
0.2812 0.3118 0.2922
0.6
Acknowledgments
0.5 0.4
The authors would like to acknowledge the financial support of the Bogazici University Research Fund with the project number 09HA203D, the TUBITAK with the project number 107E284.
0.3
References
0.1 T1FNS T2FNS Ref
0
0
1
2
3
4
5
Time (s)
ca
n.
Fig. 11. The speed responses of the motor for type-1 FNS and type-2 FNS.
[1] P. Claudia, S. Miguel, Speed control of a DC motor by using fuzzy variable structure controller, in: Proceedings of the IEEE 2008 27th Chinese Control Conference, Kunming, Yunnan, China, 2008, pp. 311–315. [2] L.A. Zadeh, Toward extended fuzzy logic–a first step, Fuzzy Sets and Systems (2009) 3175–3181. [3] Y. Tipsuwan, S. Aiemchareon, A neuro-fuzzy network-based controller for DC motor speed control, in: Proceedings of the IEEE Industrial Electronics SocietyIECON 2005, Bangkok, Thailand, 2005, pp. 2433–2438. [4] A. Niasar, A. Vahedi, H. Moghbelli, Speed control of a brushless DC motor drive via adaptive neuro-fuzzy controller based on emotional learning algorithm, in: Proceedings of the IEEE Electrical Machines and Systems, 2005-ICEMS 2005, Tehran, Iran, 2005, pp. 230–234. [5] AMIRACorp., DR300 laboratory setup speed control with variable load, Tech. Rep., AMIRA, 2000. [6] R.H. Abiyev, O. Kaynak, Fuzzy wavelet neural networks for identification and control of dynamic plants–a novel structure and a comparative study, IEEE Transactions on Industrial Electronics 55 (2008) 3133–3140. [7] M.O. Efe, O. Kaynak, B.M. Wilamowski, Stable training of computationally intelligent systems by using variable structure systems technique, IEEE Transactions on Industrial Electronics 47 (2000) 487–496. [8] A. Topalov, E. Kayacan, Y. Oniz, O. Kaynak, Adaptive neuro-fuzzy control with sliding mode learning algorithm: application to antilock braking system, in: Proceedings of the Seventh Asian Control Conference, Hong Kong, China, 2009, pp. 784–789. [9] A.V. Topalov, G.L. Cascella, V. Giordano, F. Cupertino, O. Kaynak, Sliding mode neuro-adaptive control of electric drives, IEEE Transactions on Industrial Electronics 54 (2007) 671–679. [10] N.N. Karnik, J.M. Mendel, Q. Liang, Type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems 7 (1999) 643–658. [11] J.M. Mendel, R. John, F. Liu, Interval type-2 fuzzy logic systems made simple, IEEE Transactions on Fuzzy Systems 14 (2006) 808–821. [12] C. Wang, C. Cheng, T. Lee, Dynamical optimal training for interval type-2 fuzzy neural network, IEEE Transaction on Systems, Man, and Cybernetics 34 (2004) 1462–1477. [13] H. Hagras, Comments on dynamical optimal training for interval type-2 fuzzy neural network (T2FNN), IEEE Transaction on Systems, Man, and Cybernetics 36 (2006) 1206–1209. [14] C.-H. Lee, J.-L. Hong, Y.-C. Lin, W.-Y. Lai, Type-2 fuzzy neural network systems and learning, International Journal of Computational Cognition 1 (2003) 79–90. [15] Y.-C. Lin, C.-H. Lee, System identification and adaptive filter using a novel fuzzy neuro system, International Journal of Computational Cognition 5 (2007) 15–26. [16] O. Castillo, P. Melin, Intelligent systems with interval type-2 fuzzy logic, International Journal of Innovative Computing, Information and Control 4 (2008) 771–783. [17] G. Mendez, A. Cavazos, L. Leduc, R. Soto, Modeling of a hot strip temperature using hybrid learning for interval type-1 and type-2 non-singleton type-2 FLS, in: Proceedings of the IASTED International Conference on Artificial Intelligence and Applications, Benalmadena, Spain, 2003, pp. 529–533. [18] J. Mendel, Computing derivatives in interval type-2 fuzzy logic systems, IEEE Transactions on Fuzzy Systems 12 (2004) 84–98. [19] G.M. Mendez, O. Castillo, Interval type-2 TSK fuzzy logic systems using hybrid learning algorithm, in: Proceedings of the IEEE Int. Conf. Fuzzy Systems, 2005, pp. 230–235. [20] C.-F. Juang, Y.-W. Tsao, A self-evolving interval type-2 fuzzy neural network with online structure and parameter learning, IEEE Transactions on Fuzzy Systems 16 (2008) 1411–1424. [21] X. Chen, Y. Li, R. Harrison, Y.Q. Zhang, Type 2 fuzzy logic based classifier fusion for support vector machines, Applied Soft Computing 3 (2008) 1222–1231. [22] O. Castillo, L. Aguilar, N. Cazarez, S. Cardenas, Systematic design of a stable type-2 fuzzy logic controller, Applied Soft Computing 3 (2008) 1274–1279. [23] F.-J. Lin, P.-H. Shieh, Y.-C. Hung, An intelligent control for linear ultrasonic motor using interval type-2 fuzzy neural network, IET Electric Power Applications 2 (2008) 32–41. [24] O. Uncu, I.B. Turksen, Discrete interval type 2 fuzzy system models using uncertainty in learning parameters, IEEE Transactions on Fuzzy Systems 15 (2007) 90–106.
m
0.2
−0.1
ya
0.6 0.5
ka
0.4
al
0.3
rd
0.2
.e
0.1 0 −0.1 1
2
3
w
0
w
Tacho Speed (Normalized Bipolar)
−0.3241 0.6474 1.6992
co
Tacho Speed (Normalized Bipolar)
−0.3006 0.0875 0.3021
4
T1FNS T2FNS Ref
5
w
Time (s)
Fig. 12. The speed responses of the motor for type-1 FNS and type-2 FNS.
7. Conclusion In this study a type-2 FNS is proposed for the identification and the control of a servo system with time-varying and nonlinear load conditions. The structure of the type-2 FNS is presented and the parameter update rules of the structure are derived based on fuzzy clustering and gradient descent algorithms. In order to improve the performance of the type-2 fuzzy structure, a fuzzy c-means clustering algorithm is used to determine the initial positions of the membership functions. The proposed control algorithm is tested on a real time DC motor system under different nonlinear load conditions. The experimental results obtained indicate the potential of the proposed type-2 FNS structure. It is seen that type-2 fuzzy control algorithm can handle the uncertainties in the system effectively with a much better transient performance and smaller RMSE values compared to a conventional type-1 FNS controller.
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identification and channel equalization, Applied Soft Computing 11 (2011) 1396–1406. N. Kasabov, DENFIS: dynamic evolving neural-fuzzy inference system and its application for time-series, IEEE Transactions on Fuzzy Systems 10 (2002) 144–154. R. Yager, D. Filev, Generation of fuzzy rules by mountain clustering, Journal of Intelligent & Fuzzy Systems 2 (1994) 209–219. J. Wang, A course in fuzzy systems and control, Prentice Hall, Upper Saddle River, 1997. L. Wang, C. Wei, Approximation accuracy of some neuro-fuzzy systems, IEEE Transactions on Fuzzy Systems 8 (2000) 470–478. R. Abiyev, Fuzzy wavelet neural network for prediction of electricity consumption, AIEDAM: Artificial Intelligence for Engineering Design 23 (2009) 109–118. R. Abiyev, O. Kaynak, Identification and control of dynamic plants using fuzzy wavelet neural networks, in: IEEE Multi-conference on Systems and Control, San Antonio, TX, 2008, pp. 1295–1301.
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w
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.e
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n.
co
m
[25] C. Hwang, F.C.-H. Rhee, Uncertain fuzzy clustering: interval type-2 fuzzy approach to c-means, IEEE Transactions on Fuzzy Systems 15 (2007) 107–120. [26] J.M. Mendel, Uncertain Rule-based Fuzzy Logic System: Introduction and New Directions, Prentice Hall, Upper Saddle River, 2001. [27] M.B. Begian, W.W. Melek, J.M. Mendel, Parametric design of stable type-2 TSK fuzzy systems, in: Proceedings of the North American Fuzzy Information Processing Systems, 2008, pp. 1–6. [28] M. Biglarbegian, W. Melek, J. Mendel, On the stability of interval type-2 TSK fuzzy logic control systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics 40 (3) (2010) 798–818. [29] T. Chen, Y.-C. Lin, A fuzzy back-propagation-network ensemble with example classification for lot output time prediction in a wafer fab, Applied Soft Computing 2 (2009) 658–666. [30] R.H. Abiyev, O. Kaynak, T. Alshanableh, F. Mamedov, A type-2 neuro-fuzzy system based on clustering and gradient techniques applied to system
Contents lists available at ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
A servo system control with time-varying and nonlinear load conditions using type-2 TSK fuzzy neural system Erdal Kayacan a,∗ , Yesim Oniz a , Ayse Cisel Aras a , Okyay Kaynak a , Rahib Abiyev b b
Bogazici University, Department of Electric and Electronics Engineering, 34342, Bebek, Istanbul, Turkey Near East University, Department of Computer Engineering, Mersin, Turkey
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 25 November 2010 Received in revised form 7 March 2011 Accepted 11 March 2011 Available online 4 April 2011
m
a
ca
n.
co
A type-2 Takagi-Sugeno-Kang fuzzy neural system is proposed and its parameter update rules are derived using fuzzy clustering and gradient learning algorithms. The proposed type-2 fuzzy neural system is used for the control and the identification of a real-time servo system. Fuzzy c-means clustering algorithm is used to determine the initial places of the membership functions to ensure that the gradient descent algorithm used afterwards converges in a shorter time. A number of different load conditions including nonlinear and time-varying ones are used to investigate the performance of the proposed control algorithm. The control structure has the ability to regulate the servo system with reduced oscillations when compared with the results of its type-1 counterpart around the set point signal in the presence of load disturbances. © 2011 Elsevier B.V. All rights reserved.
ka
ya
Keywords: Type-2 fuzzy neural system Gradient-based learning algorithm Fuzzy c-means clustering Servo system
al
1. Introduction
w
w
w
.e
rd
DC motors are often used in industrial control applications where speed control is required over a wide range. Moreover, they can deliver three or more times their rated torque momentarily, and supply over five times rated torque in emergency situations. One of the important features of DC motors is that the speed of the motor can be controlled smoothly down to zero. Moreover, they can respond quickly to changes in control signals due to their high ratio of torque to inertia [1]. If the parameters of a DC motor can be obtained precisely, then its control would be a relatively straightforward problem and model-based approaches (PID, pole placement, etc.) could be used. However, in much of the real world the information we have is, or is allowed to be, imprecise, uncertain, incomplete, unreliable, partially true or partially possible [2]. In general, noise in the measurements and the changes in the parameters due to the current and temperature variations give rise to uncertainties. Furthermore, the load characteristics of the DC motor are often nonlinear, such as in centrifugal fans, pumps and blowers. In such cases, not only does the performance of the model-based approaches drastically decrease, but also the complexity of the controller design increases.
∗ Corresponding author. E-mail addresses: [email protected], [email protected] (E. Kayacan), [email protected] (Y. Oniz), [email protected] (A.C. Aras), [email protected] (O. Kaynak), [email protected] (R. Abiyev). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.03.008
There are many situations in industrial control systems when control engineers face the difficulty of incomplete or insufficient information. In such cases, model-free approaches are preferred. The most common approaches to model-free design in the literature are artificial neural networks (ANNs) and fuzzy logic systems (FLSs). When the learning capability of ANNs is combined with the uncertainty-handling ability of the FLSs, the resulting structure is called fuzzy neural networks (FNNs). The application of FNNs for constructing a DC motor speed controller allows an increase in the performance of the control system [3,4]. The structures of the FLSs mentioned above can be based on TSK or Mamdani-type fuzzy reasoning mechanisms. The membership functions used in the antecedent and the consequent parts (in the case of Mamdani types) are generally type-1 that are designed and trained from numeric data sets or linguistic information. However, when there are uncertainties associated with information in the knowledge base of the process, then type-2 fuzzy sets can be utilized to handle the effects of the uncertainties in the rule base. In this study, a type-2 FNS structure is developed for such a purpose and its efficacy is verified by experimental studies on the setup DR300 AMIRA [5], by means of which time-varying loads can be generated with any non-linearity at will. One of the most important problems in the design of type-2 FNS is the learning of the system parameters. In this paper, the learning rules are derived for the adaptation of the controller parameters using fuzzy clustering and gradient algorithms. The knowledge base of the designed structures is based on type-2 TSK fuzzy rules, the antecedent parts of which are characterized by type-2 fuzzy sets, and the consequent parts use crisp linear functions. The paper is organized as
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follows: In the following section, a literature review is given. In Section 3, the mathematical description of the DC servo system is presented. In Section 4, type-2 fuzzy sets and the inference mechanism of the type-2 fuzzy TSK system are briefly introduced. In Section 5, the parameter update rules based on the clustering and the gradient algorithms are derived. In Section 6, the simulation and the experimental studies are presented for both the control and the identification of the DC motor system in the presence of load disturbances. Finally, in Section 7, conclusions are given.
Fig. 1. A servo system setup.
2. Literature review
Fig. 2. Block diagram of the motor with load.
Description
Armature terminal voltage Induced electromotive force Armature current Armature winding resistance Armature winding inductance Motor constant Magnetic excitation Speed of the rotor Load torque Acceleration torque Torque produced by the motor Moment of inertia of the motor Electrical time constant Mechanical time constant
ca
n.
UA E IA RA LA C ω ML MB M J TA TM
co
Name
m
Table 1 Nomenclature.
w
w
w
.e
rd
al
ka
ya
The combination of fuzzy systems and neural networks has recently become a popular approach in engineering fields to solve control, identification, prediction, pattern recognition and etc. One well known structure is the adaptive neuro-fuzzy inference system (ANFIS). Others include TSK-type recurrent fuzzy networks and fuzzy wavelet neural networks [6] that have been widely used for identification and control purposes. In [7], a variable structure system theory based training procedure is proposed and in [8] its use in a neuro-adaptive scheme for the control of an anti-lock braking systems is described. A fuzzy sliding mode controller with a compensator is presented as a robust control scheme in [9]. A compensator based on the sliding mode control theory is used to improve the dynamical characteristics of the drive system. Type-2 fuzzy sets have been introduced by Zadeh as the extension of type-1 fuzzy sets. Mendel and Karnik have developed the theory of type-2 fuzzy sets further in [10]. The theoretical background of interval type-2 fuzzy system and its design principles are described in [11]. Some applications of the type-2 FNSs can be seen in the literature. A recent work [12] describes the design of a type-2 FNN. The optimal training algorithm derived in that paper is later corrected in [13]. In [14] the design of a type-2 FNN structure for modeling nonlinear systems is presented. This is further developed in [15] for nonlinear system identification and nonlinear time varying channel equalization. In [13–15], type-2 Gaussian membership functions with uncertain means and fixed standard deviations are used in the antecedent part. The consequent parts of these systems include weighting interval set with unity membership grade, i.e. an interval type-1 fuzzy set. The use of interval type-2 fuzzy logic for minimizing the effects of uncertainty produced by the instrumentation elements, environmental noise and other external disturbances is discussed in [16]. In [17] the learning problems of type-2 fuzzy system and its use for controlling Hot Strip Temperature are considered. In [18] the design principles and learning of type-2 fuzzy systems are described. The type-2 FNN models used in [19,20] use type-2 fuzzy set with uncertain mean in antecedent part and type-1 interval TSK fuzzy sets in consequent parts. In [21] interval type-2 fuzzy sets are used to construct a type2 support vector machines fusion fuzzy logic system. In [22] the stability type-2 fuzzy control system is analyzed. Lin et al. [23] uses type-2 fuzzy set with uncertain mean in antecedent part and type1 interval fuzzy sets in consequent parts. Uncu and Turksen [24] proposes discrete interval valued type-2 fuzzy system models generated by a learning parameter of fuzzy c-means clustering. Hwang and Rhee [25] proposes type-2 fuzzy approach to clustering. In this paper, we use a type-2 fuzzy neuro system for DC motor control. Hybrid algorithm, which uses clustering and gradient techniques, is applied to learn the parameters of the fuzzy neuro system. Experiments have been done to demonstrate that the type-2 fuzzy neuro system can achieve robust characteristics and regulation performance with time-varying and nonlinear load conditions.
3. Mathematical description of the permanently excited DC motor The experimental setup used in this study [5] consists of two DC motors, which are connected by a mechanical clutch. The first motor is used for the control of the rotation speed or the shaft angle. The second one acts as a generator, by means of which nonlinear load conditions can be created (see Fig. 1). The block diagram of the servo system is shown in Fig. 2, the nomenclature of the symbols used being given in Table 1. From the block diagram of the system depicted in Fig. 2, the transfer function of the overall system can readily be derived as follows: ω(s) =
1 RA 1 1 + TA s UA (s) − ML (s) C 1 + TM s + TM TA s2 KM C 1 + TM s + TM TA s2
(1)
where TM =
JRA KM C
and
TA =
LA RA
The numerical values used in this study are: armature terminal voltage= 24 V; rated torque= 0.096 Nm; moment of inertia of the system= 80.45 × 10−6 kg m2 ; armature inductance= 3 mH; armature resistance= 3.13 ; back emf constant= 0.06 Vs; and torque constant= 0.06 Nm/A. 4. Type-2 fuzzy neural system structure Type-2 fuzzy systems are characterized by fuzzy IF-THEN rules. The parameters in the antecedent and the consequent parts of the rules include type-2 fuzzy values. In Gaussian type-2 fuzzy sets
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a
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b
μ(x)
μ(x) μ(x) μ(x) σ1
σ
2
c1
c2
Fig. 3. Type-2 fuzzy set with uncertain STD (a) and uncertain mean (b).
(2)
˜ and x is à ˜ and . . . and x is à ˜ IF x1 is à m 2 j1 j2 jm
w
w
w
.e
rd
al
ka
ya
where c and are the center and the width of the membership function, and x is the input vector. In this study membership functions with uncertain mean c ∈ [c1 , c2 ] are considered. The kernel of a type-2 fuzzy inference system is the fuzzy knowledge base. In a fuzzy knowledge base, the information that consists of input–output data points of the system is interpreted into linguistically interpretable fuzzy rules in IF-THEN form. In a type-2 fuzzy rule, both sides, i.e. the antecedent and the consequent parts, can be type-2 or just the antecedent part can be type-2. Mendel and his co-authors classify type-2 TSK fuzzy rules into three, depending on type-1 and type-2 nature of the antecedent membership functions and the parameters of the consequent part. In Model I, the membership functions used in the antecedent part are type-2, whereas the parameters of the consequent part are type-1 fuzzy sets. In many research studies the consequent part of (3) is taken as a type-1 fuzzy set [10,14,18–20,23]. In Model II, the membership functions used in the antecedent part are type-2, and the parameters of the consequent part are crisp numbers. In Model III, the membership functions used in the antecedent part are type-1, and the parameters of the consequent part are type-1 fuzzy sets (it is to be noted that this is not the conventional Mamdani type system as the weighted average of the inference output of each rule is taken as the output of the FLS, and this is a type-1 set, not a crisp number). This paper focuses on Model II.
m
1 (x − c)2 2 2
co
−
n.
(x) ˜ = exp
When the number of rules in a type-1 and a type-2 FNS are the same, then the number of design parameters in the latter system is more than the one in the former. The larger number of parameters increases the approximation capability of the type-2 FNS. When system data has a high level of noise and the behavior of the system is characterized by uncertainties, then a type-2 fuzzy system allows to model such systems with more accuracy than a type-1 fuzzy system. An increase in the computational load (as compared to type-1) may not be the case as a type-2 FNS may be able to describe the process using less number of fuzzy rules than a type-1 fuzzy system. In order to reduce the complexity, most researchers prefer to use interval type-2 fuzzy sets in designing type-2 fuzzy systems. Since the secondary memberships of the interval type-2 fuzzy systems are equal to one, the computations associated with interval type2 fuzzy sets are relatively manageable [26]. This is the approach followed in this paper too. Additionally, in order to further reduce the computational complexity, type-2 fuzzy sets are used in the antecedent parts of the rules and linear functions in the consequent part of the rules. The use of such a structure significantly simplifies the type reduction process as compared to the ordinary type-2 fuzzy systems. The rules used in this paper are such that the consequent parts are TSK type as indicated below:
ca
uncertainties can be associated with the mean and the standard deviation (STD). In Fig. 3(a) and (b) Gaussian type-2 fuzzy sets with uncertain STD and uncertain mean are shown. The mathematical expression for the membership function is expressed as:
THEN yj =
m
wij xi + bj
i=1
Fig. 4. Structure of type-2 TSK fuzzy neural system.
(3)
E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
i
i
Ãi (xk ) = [ i (xk ), ¯ Ãi (xk )] = [ , ¯ ] Ã
k
(4)
k
u=
(5)
w
2
w
1
.e
rd
al
In the first layer of Fig. 4, the input signals are fed to the system. In the second layer each node corresponds to one linguistic term. In this layer, for each input signal entering the system, the membership degrees and ¯ are determined using (2). The third layer calculates the firing strengths of the rules which are realized using the product t-norm operator. f = (x1 ) × (x2 ) × . . . × (xn ) Ã1 Ã2 Ãn f¯ = ¯ Ã (x1 ) × ¯ Ã (x2 ) × . . . × ¯ Ãn (xn )
w
The fourth layer determines the outputs of the linear functions yi (i = 1, . . ., n), in the consequent parts.
yj =
m
wij xi + bj
(6)
i=1
The fifth, the sixth and the seventh layers perform the type reduction and the defuzzification operations. After determining the firing strengths of rules, the defuzzified output of the type-2 TSK fuzzy system is determined. The inference engine of type-2 TSK FNS is proposed in [27,28]. The final output of type-2 FNS is determined as:
u(x) = [uL (x), uR (x)] =
···
f1 ∈ [f ,f1 ] 1
1
f1 ∈ [f ,f1 ]
f j yj
j=1 N
N i=1
fi (x)yi /
N
fi (x)
(7)
i=1
1
where uL (x) and uR (x) can be computed using iterative Karnik–Mendel algorithm [26]. However the application of Karnik–Mendel algorithm may not be convenient for the control structure of a real time implementation. For this reason [28]
q +
fj
N
f¯j yj
j=1 N
j=1
(8) f¯j
j=1
where p and q are the design parameters that weight the sharing of lower and upper firing levels of each fired rule, N is the number of active rules. f¯j and f j are determined using (5), and yj is deter-
co
m
mined using (6). These parameters can be tuned during the design of the TSK system. For example, [29] uses the least mean square method for finding the values of p and q parameters. In Fig. 4, layer 5 computes the product of the membership degrees f and f¯ and linear functions yj . Layer 6 includes two summation blocks. One of these blocks computes the sum of the output signals from the layer 5 (nominator part of (8)) and the other block computes the sum of the output signal of the layer 4 (denominator part of (8)). Layer 7 calculates output of the network using (8). After the calculation of the output signal in the type-2 FNS, the training of the network is started. The training includes an adjustment of the parameters c1ij , c2ij and ij in the membership functions in the second layer and the parameters wij , bj (i = 1, . . ., m, j = 1, . . ., n) of the linear functions in the fourth layer. In the next section, the parameter update rules of type-2 FNS are derived. 5. Parameter update rules The design of type-2 FNS includes the determination of the unknown parameters in the antecedent and the consequent parts of the fuzzy IF-THEN rules. The parameter update rules are derived for a type-2 TSK FNS with fixed mean and uncertain STD. In this paper fuzzy clustering technique is applied for the determination of the parameters in the antecedent part for modeling purposes and the gradient algorithm is used for the learning of the parameters in the antecedent and the consequent part for modeling and control purposes. In the antecedent part, the input space is divided into a set of fuzzy regions, and in the consequent part the system behavior in those regions is described. Recently a number of different approaches have been used for designing fuzzy IF-THEN rules based on clustering [25,30–32], table look-up scheme [33,34], least-squares method (LSM) [14,27], recursive least-squares (RLS) method [17], gradient algorithms [7,9,1,29,12–14,18–20,35], and genetic algorithms [6,35,36]. In this paper, the fuzzy clustering is applied to design the antecedent parts, and the gradient algorithm is applied to design the consequent parts of the fuzzy rules. The aim of clustering methods is to identify a certain group of data from a large data set, such that a concise representation of the behavior of the system is produced. Each cluster center can be translated into a fuzzy rule for identifying the class. Clustering has been used for type-1 fuzzy systems [31,32]. For type-2 fuzzy systems, subtractive clustering and fuzzy clustering have been developed [32]. Subtractive clustering is an unsupervised clustering method, in which the number of clusters for input data points is determined by the clustering algorithm. Sometimes the number of clusters in the input space should be determined in advance. In these cases, the supervised clustering algorithms are of primary concern. Fuzzy c-means clustering is one of them. It can efficiently be used for type-1 fuzzy systems [31] owing to its simple structure and sufficient accuracy. However there are still uncertainties in the determination of the model structure, such as the number of rules and the number of variables involved in the rule premises. When
ka
k
N
p
ya
of the i th input defined as a Gaussian membership function. The parameters wij and bj (i = 1, . . ., m, j = 1, . . ., n) are the parameters in the consequent part of the rules. The structure of the multi input–single output type-2 FNS used in this study is given in Fig. 4. In this structure the input signals to the network are the external input signals x = {x1 , . . ., xn }. The kernel of the fuzzy inference system is the fuzzy knowledge base which consists of the input–output data points interpreted into fuzzy rules. The type-2 FNS is constructed using type-2 TSK fuzzy rules which are given by (3). The development of the type-2 FNS includes the determination of the proper values of unknown coefficients in the antecedent and the consequent parts of each rule. Let us consider the design of a type-2 FNS when the input membership functions are of Gaussian and given by (2). If both c and parameters of the Gaussian function are taken to be uncertain (within certain intervals), the parameter space of the system can become very large. In this paper, only one of these parameters is assumed to be uncertain, i.e. uncertain STD and fixed mean or fixed STD and uncertain mean. It is to be noted that the fixed values are also subject to parameter adjustment. Due to the uncertainties in the antecedent part, the output of the type-2 fuzzy rules will have uncertainties. Let us first assume that the parameters of membership functions are represented by an uncertain mean and a fixed STD. Each membership function of the antecedent part is represented by an upper and a lower mem¯ bership function. These are denoted with (x) ¯ and (x), or A(x) and A(x).
proposes an alternative approach. In this paper, we use inference engine for type-2 TSK system proposed in [28], which is given as:
n.
where x1 , x2 , . . ., xm are input variables, yj (j = 1, . . ., n) are output ˜˜ denotes a type-2 membership function for j th rule variables, A ij
ca
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E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
rd
.e
w
and Jm2 =
C N
2 d2 um ij ji
(10)
i=1 j=1
2 1 d (ui − ui ) 2
∂E ∂¯cij
(13)
(14)
(15)
n.
∂u ∂f¯j ∂j ∂u ∂f j ∂ij + ∂f j ∂ ∂ij ¯ ij ∂ij ∂f¯j ∂ ij
j
j
¯ ij ∂u ∂f¯j ∂ ∂u ∂f j ∂ij + ¯j ∂ ∂f j ∂ ∂c1ij ¯ ij ∂c1ij ∂ f ij ¯ ij ∂u ∂f j ∂ij ∂u ∂f¯j ∂ + ¯ ∂f j ∂ ∂c2ij ∂ ¯ ∂c2 ∂fj ij ij ij
(16)
(17)
The parameters of the type-2 FNS can thus be updated using (12)–(14) together with (15)–(17). As mentioned above, the parameter p and q in (8) enable us to adjust the lower or the upper portions in the final output. In this paper the values of p and q are optimized during learning from an initial value of 0.5 using: p(t + 1) = p(t) −
∂E ∂p
(18)
q(t + 1) = q(t) −
∂E ∂q
(19)
where
where 1 ≤ m1 ≤ m2 ≤ ∞. Fuzzy partitioning is carried out through an iterative optimization of the objective functions (10), with the update of membership function and the cluster centers. For a detailed analysis, the reader can refer to [25]. After the design of the antecedents parts by fuzzy clustering, the gradient descent algorithm is applied to design the consequent parts of the fuzzy rules. In what follows, the parameter update rules are derived for a type-2 TSK FNS. At the first step, the output error is calculated: O
E=
j
∂E ∂E = ∂c2ij ∂u
w
i=1 j=1
w
Jm1 =
1 d2 , um ij ji
∂E ∂E = ∂c1ij ∂u
lower and the upper membership of the data point. Updating of the cluster centers are performed by the extension of interval type-2 fuzzy sets while executing fuzzy c-means algorithm. The interval type-2 fuzzy set is incorporated into fuzzy c-means clustering. The use of the fuzzifiers m1 and m2 result in two different objective functions to be minimized. C N
∂E ∂E ∂u ∂yj = ∂wij ∂u ∂yj ∂wij ∂E ∂u ∂yj ∂E = ∂bj ∂u ∂yj ∂bj
ca
(9)
where djk (dik ) denotes the distance between cluster j(k) and data point xi , C is the number of clusters and (xi ) and ¯ j (xi ) are the j
(12)
where is the learning rate, m is the number of input signals of the network (input neurons) and n is the number of rules (hidden neurons). The derivatives in (12) are determined by the following formulas:
∂E ∂E = ∂ij ∂u
k=1
k=1
∂E ∂bj
c2ij (t + 1) = c2ij (t) −
∂E ∂ij
co
1 ⎪ , otherwise ⎪ ⎪ C ⎪ ⎪ ⎪ 2/(m −1) 2 ⎪ (dik /djk ) ⎪ ⎩ k=1 ⎧ 1 1 1 ⎪ , if < ⎪ ⎪ C C C ⎪ ⎪ ⎪ 2/(m1 −1) ⎪ (dik /djk ) (dik /djk ) ⎪ ⎨ 1 ⎪ , otherwise ⎪ ⎪ C ⎪ ⎪ ⎪ 2/(m −1) 2 ⎪ (dik /djk ) ⎪ ⎩
ij (t + 1) = ij (t) −
∂E ; ∂c ij
bj (t + 1) = bj (t) −
The derivatives in (13) and (14) are determined by the following formulas:
k=1
k=1
∂E ; ∂wij
c1ij (t + 1) = c1ij (t) −
ya
¯ j (xi ) =
wij (t + 1) = wij (t) −
ka
j
k=1
al
(xi ) =
⎧ 1 1 1 ⎪ , if ≥ ⎪ ⎪ C C C ⎪ ⎪ ⎪ 2/(m1 −1) ⎪ (dik /djk ) (dik /djk ) ⎪ ⎨
The parameters wij , bj and c1ij , c2ij and ij (i = 1, . . ., m, j = 1, . . ., n) are adjusted using the following formulas:
m
fuzzy clustering is used in type-2 fuzzy systems, the computational complexity is reduced and a better distribution of the cluster centers is obtained. In this paper, the clustering technique proposed in [25] is used for structuring the premise part of the fuzzy system. It is to be noted that fuzzy c-means does not guarantee classification for a data set that contains clusters with different densities, that have different volumes with different numbers of data points. Because of the imperfect information about the data points, there exists uncertainties in the various parameters that are used for the assignment of fuzzy memberships. In [25], the representation and the management of these uncertainties are handled by varying the fuzzifier (parameter m) in the membership function. The fuzzifier m controls the amount of fuzziness in fuzzy classification. Consequently, the input data set is extended into interval type-2 fuzzy sets. For an input data point xi , the highest and the lowest memberships are defined using different fuzzy degrees m1 and m2 and the footprint of uncertainty is thus created. The primary memberships that extend data point xi by interval type-2 fuzzy sets are determined as [25]
5739
(11)
i=1
where O is number of output signals of the network (in the given case O = 1), udi and ui are the desired and the current output values of the network, respectively.
fj ∂E = (u − ud ) n ∂p
(20) fj
j=1
f¯j ∂E = (u − ud ) n ∂q
(21) f¯j
j=1
where ¯ yj − u ∂u (yj − u) ∂u ∂E =p n ; =q n = u(t) − ud (t); ∂u ∂f j ∂f¯j f¯j fj j=1
j=1
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; u¯ =
f¯j yj
j=1 n
fj
j=1
(22) f¯j
j=1
If t-norm prod operator is used, then ∂f j
N1
=
∂
ij
∂f¯j
; kj
∂ ¯ ij
k=1,k = / i
N1
=
¯ kj
(23)
k=1,k = / i
where i = 1, . . ., N1, k = 1, . . ., N1, and j = 1, . . ., N2. For t-norm min operator: if ( − ) ≥ 0, then kj
ij
∂f
j
∂
= 1, else
ij
¯ kj − ¯ ij ) ≥ 0, if (
then
∂f
j
∂
= 0,
ij
∂f¯j ∂ ¯ ij
= 1, else
∂f¯j ∂ ¯ ij
= 0;
where i = 1, . . ., N1, k = 1, . . ., N1, and j = 1, . . ., N2. Upper and lower membership functions between i th input and j th hidden neurons of layer 3 can be written as follows:
⎧ ⎨ G(c2 , , x ), x ≤ c1ij + c2ij ij ij i i 2 (x) = ij ⎩ G(c1 , , x ), x > c1ij + c2ij ij ij i i 2
ca
¯ ij (x) =
(24)
xi < c1ij c1ij ≤ xi ≤ c2ij xi > c2ij
ya
G(c1ij , ij , xi ), 1, G(c2ij , ij , xi ),
where G(cij , ij , xi ) is determined as: G(cij , ij , xi ) = exp
1 (xi − cij ) − 2 2
2
(25)
j
∂c1ij
∂ ¯ j (xi ) ∂c2ij ∂ (xi ) j
∂c2ij
∂ ¯ j (xi ) ∂ij
∂ (xi ) j
∂ij
=
=
=
=
=
⎪ ⎩ G(c1ij , ij , xi ) ⎧ 0, ⎪ ⎨ 0,
(xi − c1ij )
rd
ij
.e
⎪ ⎩ 0, ⎧ 0, ⎪ ⎨ 0,
c1ij ≤ xi ≤ c2ij xi > c2ij c1ij + c2ij xi ≤ 2 c1ij + c2ij xi > 2
w
∂ (xi )
=
ij2
,
xi < c1ij c1ij ≤ xi ≤ c2ij
(xi − c2ij ) ⎪ , xi > c2ij ⎩ G(c2ij , ij , xi ) ij2 ⎧ (x − c2 ) c1 + c2ij ⎪ ⎨ G(c2ij , ij , xi ) i 2 ij , xi ≤ ij
⎪ ⎩ 0,
(26)
w
∂c1ij
w
¯ j (xi ) ∂
⎧ (xi − c1ij ) ⎪ , xi < c1ij ⎨ G(c1ij , ij , xi ) 2
al
ij
Then
One important problem in the learning algorithms is the convergence. The convergence of the gradient descent method depends on the selection of the initial value of the learning rate. Usually, this value is selected within the interval [0, 1]. A large value of the learning rate may lead to unstable learning, while a small value of the learning rate results in a slow learning speed. In this study, an adaptive approach is used to update this parameters. That is, the learning of the type-2 FNS parameters is started with a small value of the learning rate . During learning, is increased if the value of the change of error E = E(k) − E(k + 1) is positive, and decreased if negative. This strategy ensures a stable learning for the type-2 FNS, guarantees the convergence and speeds up the learning.
ka
Define c cluster centers arbitrarily for the Xi data set for all Xi data set do Compute membership function using (9) for m = m1 and m = m2 Centre update. Compute cl minimum (left) and cr maximum (right) of centre using iterative algorithm given in [24] Perform defuzification using c = (cl + cr )/2 if cold = cthen Save c cluster centers and stop clustering end if end for Spread cluster’s centers as c1 = c − c c2 = c + c Assign cluster centers to the centers of membership functions for all membership functions do Determine width of membership functions as (i, j) = max(abs(c2 (i, j) − c2 (i, j − 1)), abs(c2 (i, j) − c2 (i, j + 1))) (where i = 1, . . . , m; j = 1, . . . , n, m is number of input signals, n is number of rules) end for Randomly generate w and b parameters of consequent part of type-2 FNS: Set maximum epoch number max ep no, learning tag for antecedent part learn antecedant to 0 or 1, and epoch number ep no = 1; Set initial values of p = 0.5, q = 0.5 while ep no ≤ max ep nodo for all input data pairs do For given input data calculate UT 2FNS output of type-2 FNS using (2-8) Calculate error as e(t) = UT 2FNS − U d ; (where U d is the desired output) if abs(e(t)) > then Update w, b, p, q parameters using (12-23). (where is an acceptable small value) end if if learn antecedant = 1then Update c1 , c2 and parameters of membership functions using (24-28) end if end for end while Save parameters of type-2 FNS
m
n
The design steps of type-2 FNS for the identification problem are given below:
co
f j yj
j=1
u=
n
n.
n
ij
xi >
(27)
2 c1ij + c2ij
⎧ (x − c1 )2 ⎪ ⎪ G(c1ij , ij , xi ) i 3 i , xi < c1ij ⎪ ⎪ ij ⎨
2
c1ij ≤ xi ≤ c2ij
0,
⎪ ⎪ (xi − c2ij )2 ⎪ ⎪ , xi > c2ij ⎩ G(c2ij , ij , xi ) ij3 ⎧ (x − c2 )2 c1 + c2ij ⎪ ⎪ G(c2ij , ij , xi ) i 3 ij , xi ≤ ij ⎨ ij
2
(x − c1 ) c1 + c2ij ⎪ ⎪ ⎩ G(c1ij , ij , xi ) i 3 ij , xi > ij ij
2
2
(28)
6. Simulation and experimental studies In real life, the systems can only be modeled with some level of certainty. The uncertainties cannot be adequately described by deterministic models and therefore conventional control approaches based on such models are unlikely to result in the required performance. Under such conditions the use of soft computing methodologies can be a valuable alternative. Most industrial plants have nonlinear characteristics and are susceptible to internal and external disturbances. Furthermore, the time-varying nature of the plants may be interpreted as the uncertainties in the plant coefficients. This type of uncertainty could be described using fuzzy sets. In this study the FNS structure described in the previous section is used as the identifier and the adaptive controller for the control of the laboratory setup.
E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
5741
Fig. 5. The block diagram of the identifier.
6.1. Neuro-fuzzy system for the identification of the dynamical plants
0.5
co
u(k)
0.4
n.
0.35
ca
0.3
0.25
0
ka
al
0.45 pu
(29)
K
i
400
600
800
1000
Samples Fig. 6. The input of the identifier.
0.3
RMSE
0.25 0.2 0.15 0.1 0.05
The inputs to the type-2 FNS based identifier are the input signal u(k), and 1-step delayed values of the real plant outputs y(k). The problem is to find the proper parameter values for the type-2 FNS structure such that the difference between the plant output y(k) and the identifier output yn (k) will be minimized for all input values u(k). The training of the type-2 FNS is carried out for 200 epochs with 1000 time steps in each epoch. As a performance criterion the rootmean-square-error (RMSE) defined in (30) is used.
K 1 2 RMSE = (xd − xi )
200
0.35
rd
.e
w
w
0.6 pu
⎪ ⎪ ⎩ 0.55 pu
if 0 < k ≤ 150 if 150 < k ≤ 300 if 300 < k ≤ 450 if 450 < k ≤ 600 if 600 < k ≤ 750
w
Reference(k) =
⎧ 0.4 pu ⎪ ⎪ ⎨ 0.5 pu
m
0.45
ya
The dynamical process of finding the input–output relations for a system is called identification in the literature. In identification, the clustering involves the determination of clusters in data space and the translation of these clusters into fuzzy rules such that the model obtained is close to the identified system. For this purpose, first the fuzzy clustering approach is applied to the input data points of the plant in order to determine the cluster centers. The obtained cluster centers are then used in order to organize the premise parts of the fuzzy rules. It is to be noted that it would be possible not to use a clustering algorithm and spread the membership functions onto the universe of discourse randomly. However, in this case, the gradient descent algorithm would require more time to find the final places of the membership functions. In this study, the fuzzy c-means clustering algorithm is used to determine the initial places of the membership functions so that the gradient descent algorithm converges in a shorter time. In order to prevent the system from a possible saturation, the input–output data set is collected in a closed loop fashion (Fig. 5) using a conventional PI controller. The coefficients of the PI controller are KP = 1 and KI = 5. The sampling period of the real-time system is equal to 0.01 s. The period of the reference signal is 750 samples, and the mathematical expression for the reference signal can be seen in (29):
(30)
i=1
where K represents the total number of the samples. Fig. 6 shows the input to the identifier. While Fig. 7 shows the RMSE values versus epoch number, which indicates a stable learning with the gradient based algorithm, Fig. 8 demonstrates the
0 0
50
100
150
200
Epoch number Fig. 7. RMSE versus epoch number.
output of the model and the real-time system. As can be seen from Fig. 8, the type-2 FNS gives accurate modeling results. Identification is performed with 4 fuzzy rules. These fuzzy rules are constructed using two clusters that are determined for each input variables. Different combinations of these clusters are considered for the organization of the premise parts of the fuzzy rules. As a result of clustering of premise parts and training of consequent parts, the four fuzzy rules are generated and the parameters of the type-2 FNS are determined. After the determination of the parameters in the antecedent part, the gradient algorithm is applied for the learning of the parameters in the antecedent and the consequent part. These are the
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E. Kayacan et al. / Applied Soft Computing 11 (2011) 5735–5744
0.7
Tacho Speed (Normalized Bipolar)
0.7 target model output
0.6
Motor Speed
0.5 0.4 0.3 0.2 0.1 0
0.6 0.5 0.4 0.3 0.2 0.1 T1FNS T2FNS Ref
0 0
−0.1 0
200
400
600
800
5
10
15
20
Time (s)
1000
Samples
Fig. 10. The speed responses of the motor for T1FNS and T2FNS.
Fig. 8. Output of the model and the real-time system.
Table 3 Square of the error values at each time step of the control algorithm. Load type 2
Load type 3
1.330 0.956
1.021 0.953
0.856 0.841
ka
Load type 1
al
Type-1 FNS Type-2 FNS
−0.1 0 0.1
−0.5 0.5 1.5
e(k)
e(k)
−0.2 0.1 0.4
−0.05 0.05 0.15
e(k)
oij
e(k)
e(k)
0.2 0.2 0.2
0.1 0.1 0.1
w
−0.3 0 0.3
e(k)
−0.3 0.7 1.7
w
e(k)
c2ij
e(k)
0.3 0.3 0.3
w
e(k)
.e
rd
Table 4 Initial values of the centers and standard deviation of the input membership functions for the control case. c1ij
m
0.3620 0.5393 0.3688 0.5498
n.
0.2811 0.4074 0.2920 0.4079
The structure of the control scheme is shown in Fig. 9. The variable y(k) is the output signal of the plant, g(k) is the set-point signal, e(k), e(k) and e(k) are the error, the change in the error and the sum of the error, respectively. D indicates differentiation operation and indicates integration operation. AMIRA DR300 DC motor experimental setup is used to test and compare the performances of the type-1 FNS and type-2 FNS controllers. The sampling time is set to 20 ms for all the experiments. The speed of the motor and the load torque are scaled to [−1, 1]. In order to determine the efficiency and the accuracy of the proposed controllers, three different types of load conditions are considered. Fig. 10 represents the speed responses of the motor for type-1 FNS and type-2 FNS controllers. The corresponding load condition starts with a value of 0.2 pu, and increases suddenly to 0.5 pu on 7th s, and then comes back to 0.2 pu on 14th s. As can be seen from Fig. 10, type-1 FNS and type-2 FNS adapt their parameters when the load on the motor changes suddenly in 7th s and in 14th s. In order to initialize wij , bj , c1ij , c2ij and ij , first the simulation model of the setup is run for 200 epochs. Fig. 11 shows the speed response of the motor under the sinusoidal load condition, i.e. ML (t) = 0.35 + 0.15sin (0.75 t). Besides, Fig. 12 shows the speed response of motor under load condition which is proportional to the square of the speed, i.e. ML (t) = 1.5(Speed)2 . This type of load corresponds to the load-torque characteristics of centrifugal fans, pumps and blowers. It can be inferred from Table 3 that type-2 FNS controller gives better performance than the type-1 FNS controller in all load conditions. In this table, load type 1, load type 2, and load type 3 correspond to the step change, sinusoidal, and proportional to the square of the speed of load torque, respectively. Tables 4 and 5 show the centers and standard deviations before and after control process, respectively.
ca
c2ij
Input 2
ya
c1ij
Input 1
6.2. Type-2 fuzzy neuro system for the control of a dynamical system
co
Table 2 The values of the centers of the input membership functions for the identification case after clustering.
parameters of the linear functions of the layer 4 in Fig. 4. The initial values of w, a and b are selected randomly within the interval [−1, 1]. Table 2 shows the centers of the type-2 membership functions after the clustering process. The number of rows shows the number of membership functions for each input.
Fig. 9. Structure of type-2 TSK fuzzy neural system.
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Table 5 Final values of the centers and standard deviation of the input membership functions for the control case. c1ij
c2ij −0.1006 0.0596 0.2550
−0.5004 0.6010 1.5050
oij
−0.2631 −0.1253 0.3543
−0.0896 −0.0215 0.1442
0.1022 −0.0959 0.1740
0.0735 0.1276 −0.0418
0.2812 0.3118 0.2922
0.6
Acknowledgments
0.5 0.4
The authors would like to acknowledge the financial support of the Bogazici University Research Fund with the project number 09HA203D, the TUBITAK with the project number 107E284.
0.3
References
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5
Time (s)
ca
n.
Fig. 11. The speed responses of the motor for type-1 FNS and type-2 FNS.
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m
0.2
−0.1
ya
0.6 0.5
ka
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al
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.e
0.1 0 −0.1 1
2
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−0.3241 0.6474 1.6992
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Tacho Speed (Normalized Bipolar)
−0.3006 0.0875 0.3021
4
T1FNS T2FNS Ref
5
w
Time (s)
Fig. 12. The speed responses of the motor for type-1 FNS and type-2 FNS.
7. Conclusion In this study a type-2 FNS is proposed for the identification and the control of a servo system with time-varying and nonlinear load conditions. The structure of the type-2 FNS is presented and the parameter update rules of the structure are derived based on fuzzy clustering and gradient descent algorithms. In order to improve the performance of the type-2 fuzzy structure, a fuzzy c-means clustering algorithm is used to determine the initial positions of the membership functions. The proposed control algorithm is tested on a real time DC motor system under different nonlinear load conditions. The experimental results obtained indicate the potential of the proposed type-2 FNS structure. It is seen that type-2 fuzzy control algorithm can handle the uncertainties in the system effectively with a much better transient performance and smaller RMSE values compared to a conventional type-1 FNS controller.
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