Fire-rule-based direct adaptive type-2 fuzzy H∞ tracking control

Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Fire-rule-based direct adaptive type-2 fuzzy HN tracking control Yongping Pan a,n, Meng Joo Er b, Daoping Huang a, Qinruo Wang c a b c

School of Automation Science and Engineering, South China University of Technology, Tianhe District, Guangzhou, China School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore Department of Automation, Guangdong University of Technology, Panyu District, Guangzhou, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 November 2010 Received in revised form 19 April 2011 Accepted 30 May 2011 Available online 2 July 2011

This paper presents a novel HN tracking-based direct adaptive fuzzy controller (HDAFC) for a class of perturbed uncertain affine nonlinear systems involving external disturbances and measurement noise. A practical interval type-2 (IT2) fuzzy logic system (FLS) is introduced to approximate the ideal control law. To eliminate the tradeoff between HN tracking performance and high gain at the control input, a modified output tracking error is introduced. Based on the proposed fired-ruledetermination algorithm, a practical average defuzzifier expressed in parameterized and closed formula is developed for the IT2 FLS. Without the restriction that the control gain function is exactly known, the IT2 HDAFC is constructed and its adaptive law is derived by virtue of the Lyapunov synthesis. To improve control performance under measurement noise, the recursive linear smoothed Newton predictor is further introduced as a delayless output filter. Simulated application of a single-link robot manipulator demonstrates the superiority of the proposed approach over the previous approach in terms of the settling time, tracking accuracy, energy consumption and smoothness of the control input. & 2011 Elsevier Ltd. All rights reserved.

Keywords: Direct adaptive control External disturbance Interval type-2 fuzzy logic Measurement noise Uncertain nonlinear system

1. Introduction For a class of uncertain affine nonlinear systems, the design methodology of the approximation-based adaptive fuzzy controller (AFC) first proposed by Wang (1993) has made great progress in recent years (Farrell and Polycarpou, 2006; French et al., 2003; Kovacic and Bogdan, 2006; Spooner et al., 2002; Wang, 1996). AFCs are usually classified into two categories, namely an indirect AFC and a direct AFC (Wang, 1996). In the indirect scheme, two fuzzy logic systems (FLSs) are used as estimation models to approximate the plant dynamics. In the direct scheme, only one FLS is applied as a controller to approximate an ideal control law. Generally speaking, an adaptive fuzzy control system includes uncertainties caused by unmodeled dynamics, fuzzy approximation errors (FAEs), external disturbances, etc., which cannot be effectively handled by the FLS and may degrade the tracking performance of the closed-loop system (Tong et al., 2000). The AFC combined with HN control technique is an effective approach for rejecting those uncertainties. In Chen et al. (1996), the HN tracking-based AFCs were first proposed in both indirect and direct schemes, which ensure that the L2-gain from the lumped

n

Corresponding author. Tel./fax: þ 86 20 87114189. E-mail addresses: [email protected] (Y. Pan), [email protected] (M.J. Er), [email protected] (D. Huang), [email protected] (Q. Wang). 0952-1976/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engappai.2011.05.016

uncertainty to the tracking error vector is equal to or less than a prescribed attenuation level. The HN indirect AFC has fewer constraints on the control gain than its direct counterpart and has been well-developed in recent years (Chang, 2000, 2001; Hamzaoui et al., 2004; Kung and Chen, 2007; Li and Tong, 2003; Lin et al., 2004; Rigatos, 2008; Salehi and Shahrokhi, 2008; Tong et al., 2000, 2004; Wang et al., 2002). Yet, the design of the HN direct AFC (HDAFC) is more difficult than the design of its indirect counterpart. The HDAFC proposed by Chen et al. (1996) was successfully applied to the control problem of servo drives in Rubaai (1999). Next, HDAFCs combined with the sliding-mode control technique were developed in Chang (2000, 2001) and Chen et al. (2009) to improve the robustness of the closed-loop system. In certain situations, state variables of the plant are unavailable in practice. State-observer-based output tracking HDAFCs were developed in Li and Tong (2003), Tong and Li (2002) and Tong et al. (2004) to overcome this problem. Recently, for the synchronization of uncertain chaotic systems, a quadratic optimal fuzzy-neural-network-based HDAFC was also proposed in Chen (2009). However, certain plant boundary functions are required to be known in Chang (2000, 2001) and Li and Tong (2003). Moreover, to the best of our knowledge, the control gain needs to be known in all previous HDAFC approaches, and the high gain at the control input is unavoidable in all previous HN tracking-based AFC approaches. Type-2 fuzzy set (T2 FS), which has the characteristic of its membership grades themselves being FSs, is the extension of the

Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

traditional type-1 (T1) FS (Karnik et al., 1999). An interval T2 (IT2) FS using uniformly distributed secondary grades is the simplification of the general T2 FS (Liang and Mendel, 2000). The IT2 FSbased fuzzy logic controller (FLC) can deal with linguistic and numerical uncertainties simultaneously; thus it has the potential to outperform its T1 counterpart in the presence of high uncertainty (Hagras, 2007). Compared with the traditional T1 FLC, the IT2 FLC has an additional type-reducer (TR), which is used to reduce the IT2 inference output to a T1 FS before the crisp value is obtained. Although the IT2 FLC has been successfully applied to many control fields (Hagras, 2007), the computational complexity of the TR makes the adaptive design of the IT2 FLC inconvenient. The most widespread Karnik–Mendel (KM)-algorithm-based center of sets (COS) TR (Mendel, 2007; Wu and Mendel, 2009) has two substantial drawbacks. One of them is that the closed formula cannot be obtained due to its iterative computation. Another is that its computational complexity would sharply increase in an adaptive system since the computation in each sample period depends on the ascending orders of centroids of consequent FSs (Liang and Mendel, 2000). On the other hand, discretization-based TRs or defuzzifiers of the IT2 FLC (Coupland and John, 2007; Greenfield et al., 2009a, 2009b) are not suitable for adaptive design since all of them cannot be expressed in parameterized formulas. Based on the universal approximation property of the IT2 FLS (Ying, 2008), several approximation-based IT2 AFCs were developed (Al-khazraji et al., 2011; Hsiao et al., 2008; Hwang et al., 2011; Lin and Chou, 2009; F.J. Lin et al., 2009; T.C. Lin et al., 2009, 2011; Lin, 2010a). IT2 HN indirect AFCs were also developed (Lin, 2010b; Lin and Roopaei, 2010). All the results of above approaches showed that IT2 controllers outperform their T1 counterparts in the presence of high uncertainties. However, all those approaches make use of the KM-algorithm-based COS TR and it has been pointed out in Pan and Huang (2011) that there are several deficiencies in Lin and Roopaei (2010). Recently, for a class of state feedback nonlinear systems, an IT2 HDAFC was developed in T.C. Lin et al. (2009). Nevertheless, this work has several limitations: (1) it can be applied only to the plant with a well-known control gain; (2) high control input gain is unavoidable while favorable HN tracking performance is achieved; (3) applied KM-algorithm-based COS TR cannot avoid aforementioned substantial deficiencies and (4) control input would inevitably result in serious chattering since the noisy signals are directly fed back into its HN compensator. Thus, the approaches of Lin (2010b), Lin and Roopaei (2010) and T.C. Lin et al. (2009) leading to such smooth control inputs under measurement noise is a surprising result that awaits a theoretical explanation. This paper focuses on a class of single-input single-output (SISO) perturbed uncertain affine nonlinear systems involving external disturbances and measurement noise without exact knowledge of dynamic functions. The objective is to develop an IT2 HDAFC such that the closed-loop system achieves HN tracking performance in the sense that all involving variables are uniformly ultimately bounded (UUB). The controller design is carried out by the following steps: first, to eliminate the tradeoff between HN tracking performance and high control input gain, a modified output tracking error is introduced; second, by using the proposed fired-rule-determination algorithm, a practical average defuzzifier expressed in parameterized and closed formula is developed for the IT2 FLS; third, without the restriction that the control gain function is exactly known, the control law is derived by virtue of the Lyapunov synthesis; finally, to improve control performance under measurement noise, the recursive linear smoothed Newton (RLSN) predictor in Valiviita et al. (1999) is further introduced as a delayless output filter. This paper is organized as follows. The problem under consideration is formulated in Section 2. The IT2 FLS and its practical

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average defuzzifier are proposed in Section 3. The design procedure of the IT2 HDAFC is given in Section 4. An illustrative example of controlling a single-link robot manipulator is shown in Section 5. Concluding remarks are drawn in Section 6.

2. Problem formulation Consider the following nth-order SISO affine nonlinear system (Chen et al., 1996): ( xðnÞ ¼ f ðxÞ þ gðxÞu þ dðtÞ ð1Þ y ¼ x þ randðtÞ ðn1Þ T ˙  A Rn is the measurable where x ¼ ½x1 ,x2 ,:::,xn T ¼ ½x, x,:::,x state vector, uAR and yAR are the input and output variables, respectively, f(x) is the continuous nonlinear driving function, g(x) is the continuous control gain function, d(t) denotes the external disturbance and rand(t) represents the stochastic mea˙ . .,yðn1Þ T and yd ¼ ½yd , y˙ d ,. . .,ydðn1Þ T , surement noise. Let y ¼ ½y, y,.

where yd denotes a bounded desired output single that has nthorder derivative. Define the output tracking error e¼ yd y and the ˙ . .,eðn1Þ T . error vector e ¼ yd y ¼ ½e1 ,e2 ,. . .,en T ¼ ½e, e,. Assumption 1. Functions f(x), g(x) and d(t) in (1) are unknown but bounded, i.e., for all x A D, there exist unknown bounds f ðxÞ, g ðxÞ, gðxÞ and d such that 9f ðxÞ9 r f ðxÞ, 0 og ðxÞ rgðxÞ rgðxÞ and 9dðtÞ9 rd hold, where compact set D*Rn is a certain controllable region. During the AFC design, to improve the tracking performance under the external disturbance, an additional HN compensator associated with an attenuation level is usually suggested to apply (Chen et al., 1996). If the prescribed attenuation level is smaller, the tracking performance is better while the control input gain is higher as the output of the HN compensator becomes larger. To avoid high control input gain, one introduces the following modified output tracking error (Yilmaz and Hurmuzlu, 2000): EðtÞ ¼ eðtÞZðtÞ

ð2Þ

where Z is designed to satisfy the following conditions: (1) to make E small enough at the onset of the motion t¼ 0 and (2) should rapidly vanish as the motion evolves at t 40. A suggested Z is given in the following exponential form:

ZðtÞ ¼ ða0 þ a1 t þ. . . þ an1 t n1 ÞexpðltÞ

ð3Þ

where aiAR ði ¼ 0,1,. . .,n1Þ are selected to satisfy condition 1 and

lAR þ is selected to satisfy condition 2. For the selections of ai and l, one can follow the methods in Yilmaz and Hurmuzlu (2000). Now, the objective of this paper is to determine an IT2 HDAFC such that the closed-loop system achieves: (1) the stability in the sense that all involving variables are UUB and (2) the HN tracking performance: Z T Z T ET Q E dt r2VL ð0Þ þ r2 w2 dt, T A ½0,1Þ ð4Þ 0

0 T

˙ . .,Eðn1Þ T , VL is the Lyapunov where E ¼ ½E1 ,E2 ,. . .,En  ¼ ½E, E,. þ function, rAR is the prescribed attenuation level, Q is a matrix satisfied Q¼ QT 40 and wAL2[0,T] represents the lump uncertainly. The terms Q, VL and w will be defined in Section 4. Remark 1. If the closed-loop system starts with the initial condition VL(0) ¼0, (4) can be rewritten to supw A L2 ½0,T ð:E:Q =:w: r r 2

where :E:Q ¼

RT 0

2

eT Q e dt and :w: ¼

from w to E is equal to or less than r.

RT 0

w2 dt, i.e., the L2-gain

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Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

3. Fuzzy approximator description 3.1. Interval type-2 fuzzy logic systems Consider the IT2 FLS with n inputs and a single output. Let Ui ¼[  Xi,Xi] (i¼1, y, n) and V¼[  Y,Y] be universes of discourse of input variables xi (i¼1, y, n) and output variable y, respecl li tively. Let F i ¼ fF~ g and G ¼ fG~ g denote fuzzy ðl ¼ 1,...,MÞ

i ðli ¼ 1,...,mÞ

partitions on Ui and V, respectively. Then, a fuzzy rule base with M¼mn rules is constructed as follows (Liang and Mendel, 2000): l1

ln

~ Rl : If x1 is F~ 1 and    and xn is F~ n then y is G

l1 ln

ð5Þ l1 ln

A G. where l ¼1, y, M, li ¼1, y, m, i¼1, y, n, l ¼l1y ln and G~ li ~ Consider the case of each antecedent F being IT2 FS with the i

l membership fumction (MF) m ~ li ðxi Þ and each consequent G~ being T1 Fi

FS with the MF m ~ l ðyÞ. Let m ~ li ðxi Þ and m ~ li ðxi Þ denote the lower MF Fi

G

l

F l ðxÞ ¼ ½f ðxÞ,f ðxÞ

ð6Þ

where 8 l > < f ðxÞ ¼ m F~ l1 ðx1 Þ  . . .  m F~ ln ðxn Þ

l¼1

yLðlÞ f LðlÞ Þ=

XN l¼1

f LðlÞ

By defining the L(l)th fuzzy basic function as

xLðlÞ ðxÞ ¼

X2N j¼1

1

Fn

l For all M fuzzy rules, each consequent T1 FS G~ along with its l ~ firing interval F constitutes an output IT2 FS B, which can be expressed as Z mB~ ðyÞ ¼ 1=b l l

b A ½3M ½f m ~ l ðyÞ,3M ½f m ~ l ðyÞ l ¼ 1 l ¼ 1 G

XN



f LðlÞ =

XN l¼1

 f LðlÞ =2N

yA ¼ f^ ðx9hÞ ¼ hT nðxÞ

l > : f ðxÞ ¼ m ~ l1 ðx1 Þ  . . .  m ~ ln ðxn Þ: F1

YAj ðxÞ ¼

(11) can be expressed as

n

1

where

Fi

li and the upper MF of F~ i , respectively. With the singleton fuzzifier and product t-norm (Wang, 1996), the firing interval Fl of the lth rule is obtained as follows: l

adaptive design. However, the number of total rules M is usually large, especially in the FLSs with high dimension. Consequently, the 2M in (10), which represents the total number of summation terms, is astronomical. To reduce the computational complexity of (10), the fired-rule-determination algorithm is proposed as follows: Step 1. Obtain the fired set labels Li(mi) of each antecedent IT2 li FS F~ i by the pseudo-code in Table 1, where i¼1, y, n and li ¼1, y, m. Note that the final value of mi stands for the total number of fired sets with respect to xi. Step 2. Obtain the total number of fired rules N, the labels of fired rules L(l), l ¼1, y, N, and their corresponding fired intervals FL(l) by the pseudo-code in Table 2. Consequently, with only fired rules, the crisp output of the IT2 FLS is obtained as X2N Y j ðxÞ=2N ð11Þ yA ¼ j¼1 A

G

Based on the COS TR, a T1 type-reduced set in the intervalweighted-average form is obtained as XM XM yl F l ðxÞ= F l ðxÞ ð7Þ YCOS ðxÞ ¼ l¼1 l¼1 l

where yl denotes the centroid of G~ . The left endpoint yl(x) and right endpoint yr(x) of (7) can be found by the KM algorithm (Mendel, 2007). Hence, with the centroid defuzzifier, the crisp output of the IT2 FLS is obtained as follows: yCOS ¼ ðyl ðxÞ þ yr ðxÞÞ=2

ð8Þ

ð12Þ M T

M

where h ¼ ½y ,    ,y  A R is an adaptive parameter vector, n(x)¼[n1(x), y, nM(x)]TARM is a regressive vector, and xL(l)(x) (l ¼1, y, N) are included in n(x). Lemma 1. (Ying, 2008). The IT2 FLS in (12) has the universal approximation property, i.e., for any given real continuous function f(x) on the compact set D and arbitrarily small constant mf 40, there exists a IT2 FLS in the form of (12) such that supx A D 9f ðxÞf^ ðx9hÞ9 o mf . Table 1 Pseudo-code for finding labels of fired sets. Find fired sets (n, m, x) { for (i ¼ 1; ir n; i þ þ ) { mi ¼ 0; } for (i ¼ 1; i r n; iþ þ ) { for (li ¼ 1; ir m; li þ þ ) { if m ~ li ðxi Þa 0 then {mi ¼ mi þ 1; Li ðmi Þ ¼ li ;} Fi

}

3.2. Fired-rule-based average defuzzifier Due to the iterative form of the KM algorithm, the centroid defuzzifier in (8) cannot be expressed in a closed formula. This factor brings much inconvenience for the adaptive design of the l l IT2 FLS. One observes that f and f have a total of 2M different combinations, which can be enumerated as XM XM YAj ðxÞ ¼ yl f l = fl ð9Þ l¼1 l¼1 l

l

l

M

where f is either f or f , and j¼1, y, 2 . Hence, with an average defuzzifier (Ying, 2008), the crisp output of the IT2 FLS is directly obtained as X2M yA ¼ Y j ðxÞ=2M ð10Þ j¼1 A As mentioned in Du and Ying (2010), this average defuzzifier generates an approximation of (8) with a very small approximate error. To our knowledge, (10) is the only defuzzifier of the IT2 FLS expressed in parameterized and closed formula that facilitates the

} return Li ðjÞðj ¼ 1,    ,mi ,i ¼ 1,    ,nÞ; }

Table 2 Pseudo-code for finding labels of fired rules. Find fired rules (Li ðjÞ,j ¼ 1,. . .,mi , i ¼ 1,. . .,n) { k¼0 ; for (l1 ¼ L1 ð1Þ;l1 r L1 ðm1 Þ; l1 þ þ ) { for (l2 ¼ L2 ð1Þ;l2 r L2 ðm2 Þ; l2 þ þ ) {y for (ln ¼ Ln ð1Þ;ln r Ln ðmn Þ; ln þ þ ) { k ¼ k þ 1; LðkÞ ¼ ln þ mðln1 1Þ þ    þ mðn1Þ ðl1 1Þ ; calculate F LðkÞ by (6);} } } N ¼ k; return LðlÞ and F LðlÞ ðl ¼ 1,    ,NÞ;

Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

Remark 2. Assume that the input fuzzy partitions F i ði ¼ 1,. . .,nÞ are normal, consistent and complete (Wang, 1996). Then, one has mi ¼2(i¼1,y,n) and N ¼2n. (1) As shown in the loop body of the pseudo-code in Table 1, if one uses few calculation quantities (2n times addition and mn times comparison) to find the fired sets first, the computational cost of the fired intervals could be significantly reduced since only 2N loops are needed for obtaining FL(l) with respect to fired rules compared with mN loops for obtaining Fl with respect to all rules. In summary, the first conclusion is that the fired-rule-determination algorithm is effective for reducing the computational cost of the fired intervals in the defuzzifier. (2) Since the number of fired rules N is exponentially decreased compared with the number of all rules M, the computational cost of the average defuzzifier with the fired-ruledetermination algorithm in (11) is significantly reduced compared with that of the average defuzzifier in (10). Therefore, the second conclusion is that the fired-rule-determination algorithm makes the average defuzzifier practical. 4. Controller design procedure

1177

Let h~ ¼ hhn . Noting (12) and (15), (17) becomes E˙ ¼ AE þ bgðxÞðh~ T nðxÞuh ðEÞ þ wu d=gðxÞÞ

ð18Þ

Define the lump uncertainty wL as follows (Hsueh et al., 2010): wL ¼ wu d=g þ h~ T nðxÞðg0 =g1Þ

ð19Þ

where g0 is the best estimation of g. Then, one can make the following lemma. Lemma 2. (Hsueh et al., 2010). The lump uncertainty wLALN, 8x A D, i.e., there exists a finite positive constant wL such that wL ¼ sup8x A D 9wL 9 holds. 4.2. Derivation of the control law Choose the HN compensator uh as uh ¼ ET Pb=2r2

ð20Þ

where P is the solution of the following Lyapunov equation: PA þAT P ¼ Q

4.1. Direct adaptive control scheme

ð21Þ

Choose the following Lyapunov function candidate: In the following design, the measurement noise rand(t) in (1) is not considered first. Choose a real vector k¼[kn,y,k1]T so that hðsÞ ¼ sn þk1 sn1 þ . . . þ kn is a Hurwitz polynomial, where s is a complex variable. If d ¼0 in (1), one can select the following ideal controller: un ¼

1 T ðf ðxÞ þ yðnÞ ZðnÞ þ k EÞ d gðxÞ

ð13Þ

Substitution of (13) into (1) leads to EðnÞ þ k1 Eðn1Þ þ . . . þkn E ¼ 0

ð14Þ

VL ¼ ET PE=2þ h~ T h~ =2g where gAR . Now, we state the main result of this paper.

Theorem 1. For the nonlinear system (1), select (16) that is equipped with (12) and (20) as the controller. The parameter adaptive law is designed as 8 > ð1 þ ðmu yl Þ=dÞgu ET Pbxl ðxÞ > < y˙ l ¼ ð1 þ ðmu þ yl Þ=dÞgu ET Pbxl ðxÞ > > : g ET Pbx ðxÞ u

By virtue of the selection of k, all roots of (14) are in the open left half of the s-plane. Hence limt-1 9EðtÞ9 ¼ 0, i.e., limt-1 9eðtÞ9 ¼ 0. But f and g are unknown and da0 here, thus the ideal controller in (13) cannot be realized. Define the compact sets D ¼ fx: :x:r Mx g and Ou ¼{h: 9yl9r mu þ d}, where l ¼1, y, M, yl denotes the lth element of h, Mx, muAR þ are user-defined finite constants and dAR þ is a userdefined small constant. From Lemma 1, one can let a FLS uf in the form of (12) approximate u* in (13). The minimal FAE is given by wu ¼ un uf ðx9hn Þ

ð15Þ

ð16Þ

According to (1), (13) and (16), one obtains the tracking error dynamic equation: T

EðnÞ ¼ k E þgðxÞðun uf ðx9hÞuh ðEÞÞd

E˙ ¼ AE þbðgðxÞðun uf ðx9hÞuh ðEÞÞdÞ

6 6 ^ A¼6 6 0 4 kn

^

&

0



kn1



3

7 ^ 7 7, 1 7 5 k1

˙ V˙ L ¼ ET Q E=2ET Pbg h~ T nðxÞ þ ET Pbgðwu d=gÞET Pbguh þ h~ T h~ =g ¼ ET Q E=2 þ h~ T ðh˙ =gET Pbg nðxÞÞ þ ET Pbgðwu d=gÞET Pbgu T

Noting gu ¼ gg0 and (19), (24) becomes V˙ L ¼ ET Q E=2 þ ET PbgwL ET Pbguh

ð25Þ

Substituting (20) into (25) leads to V˙ L ¼ ET Q E=2 þ gðET PbwL ðET PbÞ2 =2r2 Þ ¼ ET Q E=2gðET Pb=rrwL Þ2 =2 þg r2 w2L =2 r ET Q E=2 þ g r2 w2L =2 pffiffiffi Noting w ¼ g wL , one gets

or equivalently

0

Proof. Consider the case of ‘‘otherwise’’ in (23) first. Differentiating (22) along (18) and (23) and using (21) yield

ð24Þ

u ¼ uf ðx9hÞ þ uh ðEÞ



ð23Þ

otherwise

¼ E Q E=2 þ E Pbg h~ T nðxÞðgu =g g1Þ þ ET Pbgðwu d=gÞET Pbguh

With the additional HN compensator uh, the overall controller is designed as follows:

1

if yl o mu & ET Pb o0

where gu ¼ gg0 is a learning rate. The overall control scheme is shown in Fig. 1. Then, the closed-loop system achieves the HN tracking performance of (4) in the sense that all variables pffiffiffi are UUB, where w is a new lumped uncertainty defined as w ¼ g wL .

T

^ hÞ9Þ hn 9argminh A Ou ðsupx A D 9un uðx9

0

l

if yl 4 mu & ET Pb 40

h

where hn is an optimal parameter vector defined as

where 2

ð22Þ

þ

ð17Þ

2 3 0 6^7 6 7 b¼6 7 405 1

V˙ L rET Q E=2 þ r2 w2 =2

ð26Þ

Consider other cases in (23), one can also get (26) from the results in Chang (2001). From Lemma 2, (26) becomes V˙ L rET Q E=2 þ r2 gw2L =2

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Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

Using the same proof in Phan and Gale (2007), one gets V˙ L rkmin VL þkmin Vr where kmin ¼ lmin ðQ Þ=lmax ðPÞ, lmin(Q) is the minimal eigenvalue of Q, lmax(P) is the maximal eigenvalue of P and VrAR þ is a finite constant. Therefore, one obtains VL ðtÞ r VL ð0Þ þ Vr ,

8t Z0

where cfAR þ is the Coulomb friction torque and vfAR þ is the dynamic friction coefficient. Choose m ¼ ð2 þsinðtÞÞkg, l ¼ ð1þ 0:5sinðx1 ÞÞm, gv ¼9.8 m/s2, df ¼ 1.0 kg m2/s, cf ¼5 and vf ¼2 so that the plant is a time-varying model. Let x1 ð0Þ ¼ ðp=3Þrad and x2 ð0Þ ¼ 0rad=s.

5.2. Controller parameters design

i.e., VLALN, which implies that E, h, x, eALN. From (12), (16) and (20), one can also get uALN. Thus, all variables are UUB. Integrating (26) from t ¼0 to T yields Z Z T 1 T T 1 E Q E dt þ r2 w2 dt, T A ½0,1Þ VL ðTÞVL ð0Þ r  2 0 2 0 After some simple manipulations, the above expression can be written into (4). Remark 3. To improve the control performance of the proposed IT2 HDAFC under measurement noise, output filters should be applied. Here, we introduce the RLSN predictor with the following z-domain transfer function (Valiviita et al., 1999): P R 1 1 p i c þ SðzÞð1zp ÞMR þ z1 M i ¼ 1 HMR i ðzÞð1z Þ p ðzÞ ¼ ð27Þ HM p 1ð1cÞz where p is the prediction step, MR is the order of the predicted signal, c is the current signal gain and the low-pass filter S(z) is expressed as XN1 SðzÞ ¼ zi =N i¼0 where N is the smoothing factor. The stopband is attenuated by c, and the width of the passband is controlled by N. To avoid the noise sensitivity, MR is usually selected to be not more than two. Remark 4. As mentioned in Section 1, the previous approach of IT2 HDAFC in T.C. Lin et al. (2009) has four limitations. In the proposed approach, we derived the control law in Section 4 without knowing the exact control gain function, introduced the modified tracking error in (2) to eliminate high control input gain, developed the fired-rules-based average defuzzifier in (12) to design the IT2 HDAFC and introduced the RLSN predictor in (27) to reduce the chattering of the control input. Consequently, those limitations of IT2 HDAFC in T.C. Lin et al. (2009) can be solved or relaxed to a certain extent.

5. An illustrative example

The design procedure of the proposed controller is presented in the following steps. Step 1. Determine parameters of the modified error E in (2). Choose l ¼7 in (3). From the method in (Yilmaz and Hurmuzlu 2000), one can make  Zð0Þ ¼ a0 ¼ yd ð0Þx1 ð0Þ Z˙ ð0Þ ¼ 7a0 þa1 ¼ y˙ d ð0Þx2 ð0Þ with ym(0) ¼0, y˙ m ð0Þ ¼ p2 =2, x1(0) ¼ p/3 and x2(0) ¼0. Thus, one gets a0 ¼  p/3 and a1 ¼(p2/2 7p/3). Step 2. Construct the IT2 FLS uf in (12). Let X1 ¼X2 ¼ p. For the antecedents of uf, the MFs are designed as

mF li ðxi Þ ¼ exp½ðxi pðli 2ÞÞ2 =2ðslii Þ2  i

Adaptive law  in (23)

yd

+

E in (2)

_

Fig. 1. Overall control scheme of IT2 HDAFC.

Table 3 Performance comparisons of various controllers in Case 1. SNR

N

Defuzzifier type

KM

AVE

Consider a single-link robot manipulator governed by the following dynamic model (Feliu et al., 1993; Mei et al., 2001; Seng et al., 1998): 8 ˙ > < x 1 ¼ x2 x˙ 2 ¼ df x2 =ml2 gv cosðx1 Þ=l þ ð1=ml2 Þu þ dðtÞ ð28Þ > : y ¼ x1 þ randðtÞ where x1 is the angular position of the manipulator, x2 is the angular velocity of the manipulator, m is the mass of the payload, l is the length of the manipulator, gv is the gravitational acceleration, df is the damping factor and rand(t) represents the Gaussian white noise with SNR ¼[N,20]dB. The external disturbance d is a frictional model combined with the Coulomb friction and the viscous friction, which can be expressed as ð29Þ

x

Plant in (1)

IT2 HDAFC u in (16)

RLSN predictor H (z) in (27)

5.1. Plant dynamic model

dðtÞ ¼ ðsgnðx2 ðtÞÞcf þ vf x2 ðtÞÞ=ml2

e Modified error

40

KM

AVE 30

KM

AVE 20

KM

AVE

Controller type

Performance indexes J(ITAE)

J(IAE)

Ec

Controller Controller Controller Controller Controller

1 2 3 4 5

4.785 5.131 4.415 4.397 4.440

2.046 2.152 1.846 1.834 1.841

27,600 25,280 17,590 17,540 17,460

Controller Controller Controller Controller Controller

1 2 3 4 5

4.858 5.128 4.416 4.386 4.485

2.045 2.151 1.847 1.832 1.847

28,180 25,340 18,160 17,590 17,470

Controller Controller Controller Controller Controller

1 2 3 4 5

4.814 5.115 4.465 4.432 4.713

2.048 2.159 1.859 1.848 1.887

33,460 26,010 18,160 18,090 17,680

Controller Controller Controller Controller Controller

1 2 3 4 5

5.211 5.404 4.980 5.123 5.515

2.112 2.219 1.971 1.977 2.048

86,670 32,630 23,680 23,680 18,760

KM: KM-algorithm-based defuzzifier is applied. AVE: Average defuzzifier is applied.

Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

Rt :eðtÞ:dt, and the control energy Ec ¼ 0 u2 ðtÞdt to evaluate the control performance. To verify the effectiveness of the proposed approach, five different types of IT2 HDAFCs are applied to make the simulated comparisons, where Controller 1 is the KM-algorithm-based IT2 HDAFC (T.C. Lin et al., 2009) with the learning rate gu ¼ g, Controller 2 is obtained by removing the term g and making gu ¼g0g in Controller 1, Controller 3 is obtained by applying the modified error E in (2) to Controller 2, Controller 4 is obtained by using the average defuzzifier to replace the KM-algorithm-based defuzzifier in Controller 3 and Controller 5 is obtained by applying the RLSN predictor in (27) to Controller 4. Simulation experiments are carried out in two cases: the tracking and regulation problems.

5.3. Simulation studies Set the sample time as 1 ms and the running time as 10 s. Rt Rt 0 t:eðtÞ:dt and JðIAEÞ ¼ 0

Select the tracking indices JðITAEÞ ¼

yd x1

Angular position (rad)

2 1.5 1 0.5 0 -0.5 -1

8 Angular velocity (rad/s)

where slii A ½1:1,1:5, li ¼1,2,3 and i¼1,2. Thus there are M ¼32 ¼9 fuzzy rules for uf. For the consequents of uf, select h(0)¼[0, y, 0]T. Step 3. Select the parameters of the RLSN predictor in (27). Choose c ¼0.3, MR ¼1, N¼ 60 and p¼ 1. Step 4. Design parameters of the control law. Let k1 ¼2, k2 ¼1, Q¼diag(10,10), r ¼0.2. From (21), one gets P¼[15,5;5,5]. Let mu=300, d ¼0.2, g0 ¼1.6 and g ¼800.

2.5

-1.5 -2

y,d x2

6 4 2 0 -2 -4 -6

0

2

4

6

8

10

0

2

4

time (s)

2.5 Angular velocity (rad/s)

Angular position (rad)

1.5 1 0.5 0 -0.5 -1 -1.5 -2

8

10

y,d x2

6 4 2 0 -2 -4 -6

0

2

4

6

8

10

0

2

4

time (s)

6 time (s)

8

10

8

2.5 1.5

Angular velocity (rad/s)

yd x1

2 Angular position (rad)

6 time (s)

8 yd x1

2

1 0.5 0 -0.5 -1 -1.5 -2

1179

0

2

4

6 time (s)

8

10

y,d x2

6 4 2 0 -2 -4 -6

0

2

4

6 time (s)

8

10

Fig. 2. Response curves of Controllers 1–3 in Case 1 with SNR¼20: (a) x1 tracks yd using Controller 1, (b) x2 tracks y˙ d using Controller 1, (c) x1 tracks yd using Controller 2, (d) x2 tracks y˙ d using Controller 2, (e) x1 tracks yd using Controller 3 and (f) x2 tracks y˙ d using Controller 3.

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Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

By the use of Controller 1, acceptable tracking performance is obtained under high measurement noise (Fig. 2(a and b)). However, the control energy Ec sharply increases while the SNR value is decreased (Controller 1 in Table 3). By the use of Controller 2, the values of J(ITAE) and J(IAE) (Controller 2 in Table 3) as well as the response curves (Fig. 2(c and d)) are kept close to those of Controller 1, and much Ec is saved while the SNR value is decreased (Controller 2 in Table 3). The control performances of Controller 4 (Fig. 3(a and b), Controller 4 in Table 3) are very similar with those of Controller 3 (Fig. 2(e and f), Controller 3 in

Case 1: Tracking problem. In this case, the control adjective is to ensure that the output y tracks the desired signal yd(t)¼ (p/2)sin(pt). Performance comparisons of five types of controllers with various SNR values are shown in Table 3. Under typical measurement noise with SNR¼20 dB, response curves of the controllers using the KM-algorithm-based defuzzifier and the controllers using the average defuzzifier are shown in Figs. 2 and 3, respectively. The control inputs comparison between Controllers 4 and 5 is shown in Fig. 4. The measured and filtered state variables are depicted in Fig. 5.

2.5

Angular velocity (rad/s)

Angular position (rad)

8

yd x1

2 1.5 1 0.5 0 -0.5 -1

y,d x2

6 4 2 0 -2 -4

-1.5 -2

-6 0

2

4

6

8

10

0

2

4

2.5

8

8 Angular velocity (rad/s)

yd x1

2 Angular position (rad)

6

10

time (s)

time (s)

1.5 1 0.5 0 -0.5 -1

y,d x2

6 4 2 0 -2 -4

-1.5 -2

-6 0

2

4

6

8

10

0

2

4

time (s)

6

8

10

time (s)

Fig. 3. Response curves of Controller 4–5 in Case 1 with SNR ¼20: (a) x1 tracks yd using Controller 4, (b) x2 tracks y˙ d using Controller 4, (c) x1 tracks yd using Controller 5 and (d) x2 tracks y˙ d using Controller 5.

500

500

200

400

300

0

300

200

-100 9.8

200 9.9

10

100 50 0 -50 9.8

u

100

u

400

100

100

0

0

-100

-100

9.9

10

-200

-200 0

2

4

6 time (s)

8

10

0

2

4

6

8

time (s)

Fig. 4. Control input comparison in Case 1 with SNR ¼20: (a) control input using Controller 4 and (b) control input using Controller 5.

10

Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

Table 3), which demonstrates the effectiveness of the proposed average defuzzifier. With the modified tracking error E in (2), Controllers 3 and 4 not only eliminate high control input gain (Fig. 4(a)), but also obviously improve the tracking speed and accuracy (Figs. 2(e and f) and 3(a and b)) with much less control effort compared with those of Controller 2 (Fig. 2(c and d), Controller 2 in Table 3). The comparisons of the J(ITAE), J(IAE) and Ec values (Controllers 3 and 4 in Table 3) confirm this argument. The reason why using E in (2) can improve the transient performance is that it can eliminate temporary failure of the traditional HN compensator with e as input at the initial stage of the control action under the condition of non-zero initial state variables. However, serious chattering at the control input (Fig. 4(a)) and high energy consumption still exist under low SNR values in these controllers. By the use of Controller 5, favorable tracking performance is maintained (Fig. 3(c and d), Controller 5 in Table 3) while chattering at the control input is much reduced (Fig. 4(b)) under high measurement noise. It is shown that the influence of the RLSN predictor on the tracking performance is very small under low measurement noise, and a good tradeoff between the tracking performance and the control energy is obtained under high measurement noise. Case 2: Regulation problem. Let yc be a square signal that belongs to [ p/2, p/2] with the period equal to 10 s and the duty cycle equal to 50%. The control objective is to ensure that y tracks a filtered output yd ¼(25/s2 þ10sþ25)yc. In this case, performance comparisons of five types of controllers with various SNR values are shown in Table 4. Response curves of the controllers using the KM-algorithm-based defuzzifier and the controllers using the average defuzzifier with SNR¼20 dB are shown in Figs. 6 and 7,

Table 4 Performance comparisons of various controllers in Case 2. SNR

Defuzzifier type

N

40

30

20

Controller type

Performance indexes

0 -1

J(IAE)

Ec 4127 4713 2828 2835 2740

Controller 1 Controller 2 Controller 3

2.398 2.474 1.377

1.659 1.695 1.300

AVE

Controller 4 Controller 5

1.431 1.358

1.308 1.296

KM

Controller 1 Controller 2 Controller 3

2.452 2.594 1.377

1.666 1.708 1.300

AVE

Controller 4 Controller 5

1.426 1.374

1.307 1.299

KM

Controller 1 Controller 2 Controller 3

2.511 2.658 1.408

1.675 1.717 1.305

AVE

Controller 4 Controller 5

1.437 1.412

1.308 1.307

KM

Controller 1 Controller 2 Controller 3

3.652 3.906 3.006

1.813. 1.895 1.554

AVE

Controller 4 Controller 5

2.969 2.055

1.538 1.409

4275 4718 2823 2849 1.358 5891 5048 3053 3050 2785 27,040 18,200 12,120 11,380 3313

KM: KM-algorithm-based defuzzifier is applied. AVE: Average defuzzifier is applied.

Measured x2 Actual x2

8

1

J(ITAE) KM

10

Measured x1 Actual x1

2

respectively. The control inputs comparison between Controllers 4 and 5 is shown in Fig. 8. The measured and filtered state variables are depicted in Fig. 9. The control performances of different types of

Angular velocity (rad/s)

Angular position (rad)

3

1181

6 4 2 0 -2 -4 -6

-2

-8 0

2

4

6

8

10

0

2

4

time (s)

1.5

Angular position (rad)

4

0

Actual x1 Filtered x1

5

15 Angular velocity (rad/s)

6

1

3 2

0.5 4.6

4.7

6

8

10

time (s)

4.8

1 0

10

Actual x2 Filtered x2

-2 -4 -6 4.6

4.7

4.8

5 0

-1 -5

-2 0

2

4

6 time (s)

8

10

0

2

4

6

8

10

time (s)

Fig. 5. Measured and filtered state variables in Case 1 with SNR¼20: (a) measured and actual x1, (b) measured and actual x2, (c) actual and filtered x1 and (d) actual and filtered x2.

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Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

practical average defuzzifier expressed in parameterized and closed formula was developed for the IT2 FLS. Without the restriction that the control gain function is exactly known, the IT2 HDAFC was developed and the HN tracking performance of the closed-loop system was achieved in the sense that all involving variables are UUB. In addition, the RLSN predictor was introduced as a delayless output filter to cope with the measurement noise. From the results of the simulated application, several conclusions are obtained: (1) without the output filter, the IT2 HDAFC cannot eliminate the serious chattering at the control input under measurement noise; (2) the modified tracking error can obviously improve the transient performance with less control effort while

controllers in Case 2 are very similar with those of Case 1, which demonstrates the effectiveness of the proposed approach to deal with the regulation problem.

6. Conclusions and future works This paper has presented a novel IT2 HDAFC for a class of SISO perturbed uncertain affine nonlinear systems involving external disturbances and measurement noise. A modified output tracking error was introduced to eliminate the tradeoff between the HN tracking performance and the high control input gain. By the use of the proposed fired-rule-determination algorithm, a

2

Angular velocity (rad/s)

1.5 Angular position (rad)

4

yd x1

1 0.5 0 -0.5 -1 -1.5 -2

y,d x2

2 0 -2 -4 -6 -8

0

2

4

6

8

10

0

2

4

time (s) 2

Angular velocity (rad/s)

Angular position (rad)

8

10

4

yd x1

1.5 1 0.5 0 -0.5 -1 -1.5 -2

y,d x2

2 0 -2 -4 -6 -8

0

2

4

6

8

10

0

2

4

time (s)

6

8

10

time (s)

2

4 Angular velocity (rad/s)

yd x1

1.5 Angular position (rad)

6 time (s)

1 0.5 0 -0.5 -1 -1.5 -2

y,d x2

2 0 -2 -4 -6 -8

0

2

4

6 time (s)

8

10

0

2

4

6

8

10

time (s)

Fig. 6. Response curves of Controllers 1–3 in Case 2 with SNR ¼20: (a) x1 tracks yd using Controller 1, (b) x2 tracks y˙ d using Controller 1, (c) x1 tracks yd using Controller 2, (d) x2 tracks y˙ d using Controller 2, (e) x1 tracks yd using Controller 3 and (f) x2 tracks y˙ d using Controller 3.

Y. Pan et al. / Engineering Applications of Artificial Intelligence 24 (2011) 1174–1185

2

Angular velocity (rad/s)

Angular position (rad)

4

yd x1

1.5 1 0.5 0 -0.5 -1 -1.5

y,d x2

2 0 -2 -4 -6 -8

-2 0

2

4

6

8

10

0

2

4

time (s)

6

8

10

time (s)

2

4 Angular velocity (rad/s)

yd x1

1.5 Angular position (rad)

1183

1 0.5 0 -0.5 -1 -1.5

y,d x2

2 0 -2 -4 -6 -8

-2 0

2

4

6

8

10

0

2

4

time (s)

6

8

10

time (s)

200

200

100

100

0

0

-100

u

u

Fig. 7. Response curves of Controllers 4–5 in Case 2 with SNR¼ 20: (a) x1 tracks yd using Controller 4, (b) x2 tracks y˙ d using Controller 4, (c) x1 tracks yd using Controller 5 and (d) x2 tracks y˙ d using Controller 5.

100

-200

-100

50

-200

0

0

-300

-300 -100 9.8

-400 0

2

4

6

9.9

8

10

10

-400 0

2

time (s)

-50 9.8

9.9

6

8

4

10

10

time (s)

Fig. 8. Control input comparison in Case 2 with SNR¼20: (a) control input using Controller 4 and (b) control input using Controller 5.

eliminating the high control input gain; (3) the proposed IT2 HDAFC without using the control gain function is effective since it can also obtain favorable control performance with less control efforts; (3) the proposed average defuzzifier is effective since it can obtain similar control performance with the KM-algorithmbased defuzzifier; (5) under measurement noise, the IT2 HDAFC with the RLSN predictor is the best choice since it can significantly reduce the energy consumption and chattering at the control input. From the simulation results, one can also observe that the chattering at the control input still exists even the RLSN predictor

is applied. Further studies would focus on more effective chattering suppression methods and output feedback IT2 HDAFC under measurement noise.

Acknowledgment The first author acknowledges Prof. Jerry M. Mendel for his kind suggestion on the IT2 FLS study.

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4 Measured x1 Actual x1

2

Angular velocity (rad/s)

Angular position (rad)

3

1 0 -1 -2

Measured x2 Actual x2

2 0 -2 -4 -6 -8

-3 0

2

4

6

8

10

0

2

4

time (s)

Actual x1 Filtered x1

1 0.5 1.8

-0.5

-1.5

10

1.6

0 -1 -2 10

-3 -4

5

-5

0

-6

1.4 4.6

4.7

4.8

2

4

6

8

10

time (s)

x 10-3

-5 4.6

-7

-2 0

Actual x2 Filtered x2

1 Angular velocity (rad/s)

Angular position (rad)

1.5

-1

8

2

2

0

6 time (s)

0

4.7

2

4.8

4

6

8

10

time (s)

Fig. 9. Measured and filtered state variables in Case 2 with SNR¼ 20: (a) measured noisy x1, (b) measured noisy x2, (c) filtered x1 and (d) filtered x2.

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