914

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

Direct Model Reference Takagi–Sugeno Fuzzy Control of SISO Nonlinear Systems Mojtaba Ahmadieh Khanesar, Student Member, IEEE, Okyay Kaynak, Fellow, IEEE, and Mohammad Teshnehlab

Abstract—This study presents a novel direct model reference fuzzy controller. It relaxes the special conditions on the reference model that is required by some of the approaches described in the literature, as well as covering a more general class of Takagi– Sugeno (T–S) systems. The stability of the proposed method is proved using a proper Lyapunov function. In addition, the effects of modeling errors on the proposed controller are considered, and a robust modification algorithm to alleviate this problem is introduced and analyzed. The proposed method is then simulated on a flexible joint robot in a feedback linearization form and on Chua’s chaotic electrical circuit. Finally, it is implemented and tested on a nonlinear dc motor with nonlinear state-dependent disturbance. Index Terms—Fuzzy control, model reference adaptive control, Takagi–Sugeno (T–S) fuzzy model.

I. INTRODUCTION HE problem of controlling nonlinear systems by the use of fuzzy structures and classical control techniques is widely investigated in a number of previous studies. In some of these approaches, the fuzzy system is used as a powerful general function approximator, and different classical methods are used to estimate the parameters of the fuzzy control system. For example in [1], a fuzzy sliding mode controller is proposed, and in [2] and [3], a fuzzy-identification-based back-stepping controller and a model reference controller with an adaptive parameter estimator that is based on Takagi–Sugeno (T–S) fuzzy models are proposed, respectively. Using a model system to generate the desired response is one of the most important adaptive control schemes [4] that are studied in such hybrid approaches, and to date, different fuzzy model reference approaches have been proposed. The indirect model reference fuzzy controllers that are described in [3] and [5]–[8] and the direct model reference fuzzy controllers for T–S fuzzy models that are described in [9]–[11] can be cited as some examples. This paper proposes a novel direct model reference fuzzy controller. As the title suggests, the proposed controller does not need any identifier, and as such, the complexity and the computational cost of the controller are reduced. This is the most obvious superiority of the proposed controller, over the indirect version that is proposed in [3] and [5]–[7]. The direct model reference fuzzy controllers that are proposed in [9] and [10] re-

T

Manuscript received July 18, 2010; revised January 13, 2011; accepted March 25, 2011. Date of publication May 5, 2011; date of current version October 10, 2011. M. A. Khanesar and M. Teshnehlab are with the Department of Control Engineering, K. N. Toosi University of Technology, 19697, Tehran, Iran (e-mail: [email protected]; [email protected]). O. Kaynak is with the Department of Electrical and Electronics Engineering, Bogazici University, 34342, Istanbul, Turkey (e-mail: okyay.kaynak@ boun.edu.tr) Digital Object Identifier 10.1109/TFUZZ.2011.2150757

quire the solution of linear matrix inequalities (LMIs), which is done offline and is time consuming. In addition, it is not possible to use every reference model as desired. To solve this problem, the authors of [11] propose a new direct model reference fuzzy controller that does not need the solution of such LMIs. However, the proposed method can only control a class of fuzzy T–S models that has the same B (the gain of input) in all rules. The model reference fuzzy controller that is designed in [12] also suffers from the same problem. The direct model reference system that is studied in [13] can be cited as a further example to the work reported in the literature. However, the approach that is described therein is for a first-order nonlinear dynamical system. Some exemplary papers addressing the design of model reference fuzzy controllers are [14]–[16]. The novel direct model reference fuzzy controller that is described in this study differs from those described in the literature because 1) it is capable of controlling a more general class of T–S fuzzy systems that do not necessarily have the same gain of input in all rules, and 2) the order of the nonlinear dynamical system considered can be more than 1. The proposed method is simulated on a flexible joint robot in feedback linearization form. To show that it is not always necessary to have a system in a feedback linearization form, the proposed method is tested on Chua’s electrical circuit that is not in feedback linearization form. Finally, the proposed method is implemented and tested on a nonlinear real-time dc motor, which is loaded by a nonlinear load that is relative to the square of velocity of the motor. This type of load corresponds to the load–torque characteristics of centrifugal fans, pumps and blowers. II. DESIGN OF DIRECT MODEL REFERENCE FUZZY CONTROLLER A. Fuzzy Takagi–Sugeno Modeling The basic idea behind the fuzzy T–S modeling is to describe a nonlinear system by some fuzzy locally linear IF–THEN rules. The overall nonlinear model of the system is achieved by a fuzzy blending of the linear system models. In [17], it is proven that the T–S fuzzy models are universal approximators. The concept of T–S fuzzy model is described as follows. Consider the continuous-time nonlinear system described by the T–S model. The ith rule of the continuous-time T–S model is of the following form: Ri : IF x1 is C1i and . . . and xn is Cni THEN x˙ = Ai x + Bi u,

i = 1, . . . , m

where xT (t) = [x1 , x2 , . . . , xn ], and m is the number of rules of the T–S system. Here, it is assumed that Ai , Bi (i = 1, . . . , m)

1063-6706/$26.00 © 2011 IEEE

KHANESAR et al.: DIRECT MODEL REFERENCE TAKAGI–SUGENO FUZZY CONTROL OF SISO NONLINEAR SYSTEMS

are the state matrixes of the system and the sign of the elements is valid of Bi are the same in all rules. This assumption  for most  practical plants [5]. Given a pair of input xT (t), u(t) , the final output of the fuzzy system is inferred as follows: m wi (x(t)){Ai x(t) + Bi u(t)} m ˙ (1) x(t) = i=1 i=1 wi (x(t))  where wi (x(t)) = nj=1 Mji (x(t)) and Mji are the grades of the membership function of xj (t) in Cji . A compensator is used for each rule of the fuzzy model. The ith rule of the T–S fuzzy controller is of the following form: Ri : IF x1 is C  1 and . . . and xn is C  n i

THEN u = −Ki x + li r,

i

i = 1, . . . , m

where Ki and li , i = 1, . . . , m are the gains of the controller, and the dimension of Ki ’s, li ’s are 1 × n and 1 × 1, respectively. The computational output of the fuzzy controller is inferred as follows: m w i (x(t)) [−Ki x + li r] (2) u = i=1 m  i=1 w i (x(t))  where w i (x(t)) = wi (x(t))|li−1 | = nj=1 Nji (x(t)) and Nji are the grades of the membership functions of xj (t) in C  ij and are equal to  Nji = n |li |−1 Mji . (3)

Since the sign of li is the same for i = 1, . . . , m, one has m 

wi li−1 u −

i=1

In this section, the design of the direct model reference fuzzy controller is described. It is assumed that the system can be modeled by the T–S fuzzy model of (1). It is further assumed that the reference model is a linear system as x˙ m = Am xm + Br r.

(4)

and

Br = Bi li∗

By subtracting (6) from (4), the following is obtained: m wi Br l∗−1 K∗i x e˙ m = Am em + i=1 m i i=1 wi m m ∗−1 wi Br li i=1 i=1 wi Br r m + u−  . m i=1 wi i=1 wi The control signal is considered as m wi |li |−1 [−Ki x + li r] . u = i=1 m −1 i=1 wi |li |

(7)

(8)

(9)

i=1

  wi Br K∗i li∗−1 − Ki li−1 x m = Am em + i=1 wi  ∗−1  m − li−1 i=1 wi Br li m + u (11) i=1 wi

e˙ m

i=1

and finally m e˙ m = Am em +

˜ i l∗−1 wi Br K i i=1 m w i i=1

x

  wi Br Ki li∗−1 − li−1 m + x i=1 wi   m wi Br l∗−1 − li−1 + i=1 m i u i=1 wi m

i=1

(12)

˜ i = K∗ − Ki . To analyze the stabil˜ i is defined as K in which K i ity of the system, the following Lyapunov function is considered: m ˜ T ∗ −1 ˜ m ˜T ∗ −1 ˜ l |l | li K |l | Ki V = eTm Pem + i=1 i i + i=1 i i (13) γ1 γ2 in which ˜li = li∗ − li . The time derivative of the Lyapunov function is derived as m 2  ˜ T ∗−1 ˜˙ V˙ = eTm Pe˙ m + e˙ Tm Pem + K |l |Ki γ1 i=1 i i

(5)

so that (1) can be rewritten as m m wi Br l∗−1 K∗i x w B l∗−1 m i r i u. (6) + i=1 x˙ = Am x + i=1 m i i=1 wi i=1 wi

wi li−1 [−Ki x + li r] = 0.

m

A compensator is designed for each rule of the fuzzy system. It is assumed that there are K∗i and li∗ such that Am = Ai − Bi K∗i ,

m 

By adding (9) to (7), one obtains m m wi Br li∗−1 K∗i x wi Br li∗−1 u i=1  i=1  + e˙ m = Am em + m m i=1 wi i=1 wi m  m ∗−1 −1 ∗ wi l Br li wB l m i r i u − i=1m i r − i=1 i=1 wi i=1 wi m −1 wi li [−Ki x + li r] (10) + Br i=1  m i=1 wi

As each fuzzy rule is a linear model, linear control techniques can be used. The resulting overall controller, which is nonlinear in general, is a fuzzy blending of each individual linear controller. B. Design of the Direct Model Reference Fuzzy Controller

915

+

m 2  ˜T ∗−1 ˜˙ l |l |li γ2 i=1 i i

(14)

so that V˙ = eTm PAm em + eTm ATm Pem  m m ˜ i l∗−1 x wi Br K wi Br (l∗−1 − li−1 ) T i i=1  + 2em P + i=1 m i u m i=1 wi i=1 wi  ∗−1  m − li−1 x i=1 wi Br Ki li m + i=1 wi +

m m 2  ˜ T ∗−1 ˜˙ 2  ˜T ∗−1 ˜˙ l |l |li . Ki |li |Ki + γ1 i=1 γ2 i=1 i i

(15)

916

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

In the other case, that is, when li = l0 , the time derivative of the Lyapunov function (13) becomes m wi Br (li∗−1 − l0−1 )(Ki x + u) T T ˙ m V = −em Qem + 2em P i=1 . i=1 wi (21) By the use of the prior knowledge that |li∗ | > l0 one gets (li∗−1 − l0−1 )sign(li∗−1 ) < 0 so that m wi Br (li∗−1 − l0−1 )(Ki x + u) T m <0 (22) em P i=1 i=1 wi and this means that V˙ is negative. Consequently, in either case of (19), one obtains Fig. 1.

V˙ ≤ −eTm Qem

Block diagram of the direct model reference fuzzy controller.

so that By the use of adaptation laws as ˙ i = −K ˜˙ i = K ˙ l˙i = −˜li =

wi BT Pem xT γ1 sign(li∗ ) rm i=1 wi

wi BTr −γ2 sign(li∗ )

,

γ1 > 0

Pem (u + Ki x)  , li m i=1 wi

(16) γ2 > 0

 0

∞

eTm em ≤

V (0) − V (∞) . λm in (Q)

(23)

(24)

This means that em ∈ 2 . Furthermore, as V ∈ ∞ , one has em , Ki , li ∈ ∞ . In addition, since |li | has a lower bound, μi (x) ∈ ∞ so that e˙ m ∈ ∞ . Using Barbalat’s lemma, one has limt→∞ em (t) = 0. 

(17) D. Effect of Modeling Error and Robust Adaptive Law

one obtains V˙ = −eTm Qem .

(18)

As the adaptive gain li appears in the denominator of the fuzzy controller, zero value must be prevented. To achieve this, the adaptation law of li is modified as follows:

ϕi , if |li | > li0 or li = li0 and ϕi sign(li ) < 0 ˙li = 0, otherwise ϕi = −γ2 sign(li∗ )

BTr Pem (u + Ki x)  . li m i=1 wi

(19)

The block diagram of the resulting direct model reference fuzzy controller is shown in Fig. 1. C. Stability Analysis of the Proposed Direct Model Reference Fuzzy Controller The following theorem summarizes the stability discussions presented earlier. Theorem 1: Consider the plant model of (1) and the reference model of (4) with the control law of (2) and the adaptation laws of (16) and (19). Assume that the reference input is uniformly bounded and the reference model is stable. It is guaranteed that all the signals, including Ki (t), li (t), μi (t) and em (t) for all i = 1, . . . , m are bounded, as well as limt→∞ em (t) → 0. Proof: By the use of adaptation laws of (16) and (19), the stability of the system can be analyzed as follows. In the case when li > li0 or li = li0 and ϕi sign(li∗ ) < 0, one obtains V˙ = −eTm Qem as discussed earlier.

(20)

In the previous section, it is assumed that the only uncertainty in the dynamical system is due to unknown controller parameters. However, in practical applications, modeling errors are inevitable when describing real systems with T–S fuzzy model. The minimum functional approximation error (MFAE) is the most important modeling error [18] that appears when a nonlinear system is described with a T–S fuzzy model. Since the nonlinear equations that describe the dynamics of the nonlinear system can be in any form and the fuzzy system is constructed generally by exponential or triangular membership functions, it is practically impossible to obtain the zero modeling error. That is to say the T–S fuzzy model cannot exactly match the nonlinear system, and therefore, the system exhibits an MFAE. Furthermore, there are other sources of modeling errors, such as unmodeled dynamics, measurement noise, time variation of parameters, and disturbances [18]. Here, it is assumed that the T–S fuzzy model of (1) does not exactly describe the nonlinear system, resulting in the modeling error (MFAE) (x, u) and a stability analysis is presented under this condition. Consider the following general nonlinear function: x˙ = f (x) + g(x)u

(25)

in which f : Rn → Rn , and g : Rn → Rn are two unknown functions. Let G(x, u) = f (x) + g(x)u on the compact set X × U be an affine continuous function with G(0, 0) = 0, and f (x) be continuously differentiable on X. Further, it is assumed that X × U is a compact set on Rn × R. It can be shown that for any ε > 0, there exists a T–S fuzzy system [19], [20]: m wi (x(t)){Ai x(t) + Bi u(t)} m ˙ (26) x(t) = i=1 i=1 wi (x(t))

KHANESAR et al.: DIRECT MODEL REFERENCE TAKAGI–SUGENO FUZZY CONTROL OF SISO NONLINEAR SYSTEMS

such that

−ν

f (x) + g(x)u m wi (x(t)){[Ai + A i (x)]x(t) + [Bi + B i (x)]u(t)} m = i=1 i=1 wi (x(t))

m 

917

˜i −ν ˜ T |l∗−1 |K K i i

i=1

+ν

m 

m 

|li∗−1 |˜li2

i=1

˜ i |l∗−1 | − Ki em 2 K i

i=1

(27) +ν

and

m 

l˜i |li∗−1 | − li em 2

(33)

i=1

A i (x)∞ < ε,

B i (x)∞ < ε,

i = 1, . . . , m.

(28)

Following a similar analysis as in the previous section, one has m m wi Br li∗−1 K∗i x w B l∗−1 i=1  m i r i u + i=1 x˙ = Am x + m i=1 wi i=1 wi + (x, u).

(29)

The same Lyapunov function as in (13) is considered, and the adaptation law is modified [18], [21] as ˙ i = −K ˜˙ i = K

l˙i =

ϕi

in which

wi BT Pem xT γ1 sign(li∗ ) rm i=1 wi

if |li | > li0

− γ1 νKi em  (30)

or li = li0 and ϕi sign(li ) < 0

0

(35)

ρ = λm ax (P)εxm 2 + 2λm ax (P)ε

+ν

m 

Ki em 2

i=1

+ 2λm ax (P)εem 

[|li r| + Ki xm ]

i=1 m 

˜i −ν ˜ T |l∗−1 |K K i i

i=1 m 

m 

|li∗−1 |˜li2

i=1

˜ i |l∗−1 | − Ki em 2 K i

i=1 m 

l˜i |li∗−1 | − li em 2 .

(32)

i=1

Q can be chosen so that 6λm ax (P)ε < λm in (Q), and we have 1 V˙ ≤ − λm in (Q)em 2 + λm ax (P)εxm 2 2 m  + 2λm ax (P)ε Ki em 2 i=1

+ 2λm ax (P)εem 

m  i=1

m 

[|li r| + Ki xm ]

[|li r| + Ki xm ]

m 

˜ i |l∗−1 | − Ki em 2 K i

i=1

V˙ ≤ − λm in (Q)em 2 + 3λm ax (P)εem 2

m 

Ki em 2

i=1

wi BTr Pem (u + Ki x)  − γ2 νli em . li m i=1 wi

+ λm ax (P)εxm 2 + 2λm ax (P)ε

m  i=1

+ 2λm ax (P)εem 

For the case when |li | > li0 or li = li0 and ϕi sign(li ) < 0, the time derivative of the Lyapunov function is obtained as

+ν



min λm i n2 (Q ) , ν

 c= −1 −1 max λm ax (P), γ1 , γ2

and

(31)

+ν

(34)

otherwise

ϕi = −γ2 sign(li∗ )

−ν

V˙ ≤ −cV + ρ

+ν

m 

l˜i |li∗−1 | − li em 2 .

(36)

i=1

Therefore, the Lyapunov function V converges exponentially until V ≤ ρc , and the parameters of the controller are bounded. The region in which the system is stable is defined as

ρ ˜ i , ˜li , em ), i = 1, . . . , m . (37) R = x| < V (K c It is important to note that ε is a small positive number and theoretically it can be chosen as small as desired using sufficient number of rules for the T–S fuzzy system. A small value for ε results in a smaller ρ and bigger stable region for the system. It should also be noted that ν plays an important role in the robustness of the system in the presence of a MFAE. Since ν is the leakage parameter, a high value for ν would make the system more robust, while a small ν would make the performance of the system better. III. SIMULATION OF THE PROPOSED METHOD ON A FLEXIBLE JOINT ROBOT In this section, the proposed method is simulated on a flexible joint robot. Fig. 2, schematically, depicts the arrangement of the flexible joint robot. Define q = {q1 q˙1 q2 q˙2 } as the set of generalized coordinates for the system [22], where we have the following. 1) q2 = −(1/m)θ1 is the angular displacement of the rotor, and m is the gear ratio. 2) q1 is the angle of the link, and q1 − q2 is the elastic displacement of the link.

918

Fig. 2.

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

Single flexible joint mechanism.

Using Euler–Lagrange equations, the analytical model of the flexible joint robot is derived as [22]

I q¨1 + M gLsin(q1 ) + K(q1 − q2 ) = 0 (38) J q¨2 − K(q1 − q2 ) = u. The state-space representation of the system is derived as

Fig. 3.

Fuzzy membership functions used to model the flexible joint robot.

are considered in the simulation studies are g = 9.8 m/s2 , M = 1 kg, K = 1 N/m, J = 1 kg·m2 , I = 1 kg·m2 , and L = 1 m.

x˙ 1 = x2 x˙ 2 = −

A. T–S Fuzzy Model of the Flexible Joint Robot

M gL K sin(x1 ) − (x1 − x3 ) I I

x˙ 3 = x4 K 1 (x1 − x3 ) + u (39) J J where x1 = q1 , and x3 = q2 . The state-space form of (39) does not satisfy the conditions stated in (5). As this system is feedback linearizable, the coordinates of the system can be changed [23] to satisfy the conditions of (5). The resulting system will be in the form of x˙ 4 =

z˙1 = z2 z˙2 = z3 z˙3 = z4



 M gL k K cosz1 + + z˙4 = − z3 I I J   M gL K K u + z22 − sin (z1 ) + I J IJ

(40)

in which z1 = x1 z2 = x2 M gL K sin(x1 ) − (x1 − x3 ) I I M gL K x2 cos(x1 ) − (x2 − x4 ) (41) z4 = − I I Equation (40) indicates that the resulting system satisfies the conditions of (5). The numerical values of the parameters that z3 = −

To model the flexible joint robot, the fuzzy membership functions that are shown in Fig. 3 are considered. The parameters z1 and z2 form the rule base of the T–S fuzzy model. As z1 = x1 = q1 and z2 = x2 , these parameters of the fuzzy system are the real states of the system. The resulting T–S fuzzy model is obtained by linearizing the system at the center of the membership functions of z1 and z2 as follows. Rule 1: IF z1 IS ABOUT(0), AND z2 IS ABOUT(0) THEN z˙ = A1 z + B1 u; Rule 2: IF z1 IS ABOUT(0), AND z2 IS ABOUT(π) THEN z˙ = A2 z + B2 u; Rule 3: IF z1 IS ABOUT(0), AND z2 IS ABOUT(−π) THEN z˙ = A3 z + B3 u; Rule 4: IF z1 IS ABOUT(π), AND z2 IS ABOUT(0) THEN z˙ = A4 z + B4 u; Rule 5: IF z1 IS ABOUT(π), AND z2 IS ABOUT(π) THEN z˙ = A5 z + B5 u; Rule 6: IFz1 IS ABOUT(π) AND z2 IS ABOUT(−π) THEN z˙ = A6 z + B6 u; Rule 7: IFz1 IS ABOUT(−π), AND z2 IS ABOUT(0) THEN z˙ = A7 z + B7 u; Rule 8: IF z1 IS ABOUT(−π), AND z2 IS ABOUT(π) then z˙ = A8 z + B8 u; Rule 9: IF z1 IS ABOUT(−π), AND z2 IS ABOUT(−π) THEN z˙ = A9 z + B9 u; where ⎡ ⎤ 0 1 0 0 ⎢ 0 0 1 0⎥ ⎥ ⎢ A1 = ⎢ ⎥ ⎣ 0 0 0 1⎦ −9.8

0

−11.8

0

KHANESAR et al.: DIRECT MODEL REFERENCE TAKAGI–SUGENO FUZZY CONTROL OF SISO NONLINEAR SYSTEMS

⎡ ⎢ ⎢ A 2 = A3 = ⎢ ⎣

0

1

0

0 0

0 0

1 0

0

⎤

0⎥ ⎥ ⎥ 1⎦

0 −11.8 0 ⎤ 0 1 0 0 ⎢ 0 0 1 0⎥ ⎥ ⎢ A 4 = A7 = ⎢ ⎥ ⎣ 0 0 0 1⎦ 9.8 0 7.8 0 ⎡ 0 1 0 ⎢ 0 0 1 ⎢ A 5 = A6 = A8 = A9 = ⎢ ⎣ 0 0 0 −86.92 0 7.8 ⎡

919

86.92

B i = [ 0 0 0 1 ]T ,

⎤ 0 0⎥ ⎥ ⎥ 1⎦ 0

i = 1, . . . , 9.

Fig. 4. robot.

Regulation response of the controller when applied to the flexible joint

Fig. 5. robot.

Tracking response of the controller when applied to the flexible joint

B. Structure of the Proposed Controller Let Am of the reference model (4) be ⎡ 0 1 0 ⎢ 0 0 1 ⎢ Am = ⎢ ⎣ 0 0 0 −160

−192

−82

⎤ 0 0 ⎥ ⎥ ⎥. 1 ⎦

(42)

−15

This reference model corresponds to the desired poles of {−2, −5, −4, −4} for the system. Using this reference model and taking Q as ⎤ ⎡ 1 0 0 0 ⎢0 1 0 0⎥ ⎥ ⎢ (43) Q=⎢ ⎥ ⎣0 0 1 0⎦ 0 one has

⎡

0.9948

⎢ −0.5000 ⎢ P=⎢ ⎣ −0.9050 0.5000

0

0

1 ⎤

−0.5000

−0.9050

0.9050 −0.5000

−0.5000 3.5511

−3.5511 ⎥ ⎥ ⎥. −0.5000 ⎦

−3.5511

−0.5000

42.8880

0.5000

(44)

The premise part of the fuzzy controller is achieved using the T–S model as in the previous section and (3). The initial values of the gains of the consequent part Ki and li are found as B i K i = Ai − A m , B i li = B r ,

i = 1, . . . , 9

i = 1, . . . , 9

(45) (46)

where Ai and Bi are given in the previous subsection. The simulations are done using the Simulink software. To examine the regulation and the tracking performances of the system, the initial values of the states of the system are taken as z = [ 1 0 0 0 ]T , and those for the reference model are considered as zm = [ 1.1 0 0 0 ]T . The regulation results are shown in Fig. 4 and the tracking response for a sinusoidal reference in Fig. 5, the adaptation gains in (16) and (19) being γ1 = 0.001, γ2 = 0.1, respectively.

Figs. 4 and 5 indicate that the system behaves as the linear reference model with the adaptation laws of (16) and (19) despite the plant-model mismatches that must inevitably exist. To show the effectiveness of the terms added to the adaptation law against modeling errors, another simulation study is carried out with the adaptation laws of (30) and (31) with γ1 = 0.001, γ2 = 0.1, and ν as ν = 0.01. The Frobenius norm of K obtained is shown in Fig. 6. When the parameter ν is changed to 0, that is, when the term that ensures robustness is turned off, the resulting Frobenius norm of K is shown in Fig. 7. It can be seen that although the adaptation laws of (16) and (19) which are used here are stable and result in the performances shown in Figs. 4 and 5, because of the MFAE problem and lack of persistantly exciting control signal, the Frobenius norm of the parameter K is growing. This indicates the need for the use of the robustness term to overcome the problems caused by various modeling errors, especially the MFAE problem. It should be noted that the addition of this term does not affect the tracking performance significantly. Here, it is assumed that the control signal has no limitations and that its value can be as large as needed. However,

920

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

in which u1 , u2 , and u3 are the external inputs. Considering q = − 100 7 and u2 = u3 = 0 [27], one obtains the state-space equations of the system as   1 x˙1 = p x2 − (2x31 − x1 ) + u1 7 x˙2 = x1 − x2 + x3 100 .x2 x˙3 = − 7

Fig. 6.

in which p = 10 [27]. By the use of the T–S membership functions labeled as about(−1), about(0), and about(1) as exp(−(x1 + 1)2/0.422 ), exp(−x21 /0.422 ), and exp(−(x1 − 1)2/0.422 ), respectively, the following rules for the T–S fuzzy model are considered. Rule 1 IF x1 IS ABOUT(−1), THEN x˙ = A1 .x + B1 .u; Rule 2 IFx1 IS ABOUT(0), THEN x˙ = A2 .x + B2 .u; Rule 3 IF x1 IS ABOUT(1), THEN x˙ = A3 .x + B3 .u; in which

Frobenius norm of K when the robustness term is in effect.

⎤

⎡

5 − p p ⎢ 7 ⎢ −1 A1 = A3 = ⎢ ⎢ 1 ⎣ 100 0 − 7 ⎡ ⎤ 1 ⎢ ⎥ B1 = B2 = B3 = ⎣ 0 ⎦ . 0

Fig. 7.

0

⎡1

⎤ p

⎥ ⎢7 ⎥ ⎢ ⎥ 1 ⎥, A2 = ⎢ ⎢ 1 ⎦ ⎣ 0 0

p −1 −

100 7

0

⎥ ⎥ 1⎥ ⎥ ⎦ 0

(49)

The initial value of the chaotic system is taken as [0.707 0 0.707]. The reference model for the system is taken as

Frobenius norm of K without the robustness term.

⎡ in reality there are some limitations in the value of the control signal such as the maximum current, maximum torque, etc. This paper does not deal with saturation in the control signal. IV. SIMULATION OF THE PROPOSED DIRECT MODEL REFERENCE ADAPTIVE FUZZY CONTROL ON A MODIFIED CHUA’S CIRCUIT SYSTEM In order to show the applicability of the proposed controller on nonlinear systems that are not in feedback linearization form, the modified Chua’s circuit is considered. In the literature a number of control approaches for this circuit can be found, such as state feedback [24], nonlinear linearization [25], sliding mode control [26], nonlinear feedback control [27], etc. The modified Chua’s circuit is described by the following dynamical system [27]:   1 x˙ = p y − (2x3 − x) + u1 7 y˙ = x − y + z + u2 z˙ = −q.y + u3

(48)

(47)

−15

⎢ Am = ⎣ 1 0

−35.7

−11.5

−1

1

−14.2857

0

⎤ ⎥ ⎦.

(50)

This reference model corresponds to the eigenvalues as [−1, −5, −10]. Fig. 8 shows the regulation response of the chaotic system, and Figs 9 and 10 the tracking performance when the reference signal for x1 is a square wave and sinusoid, respectively. This example shows that although the system is not in feedback linearization form, it satisfies the conditions of (5), and it is possible to use the real states of the system for control, and no change in coordinates is needed. V. IMPLEMENTATION OF THE PROPOSED DIRECT MODEL REFERENCE ADAPTIVE FUZZY CONTROL ON A NONLINEAR DC MOTOR WITH A VARIABLE NONLINEAR LOAD To show the practical implementability of the proposed direct model reference fuzzy controller, it is used to control a nonlinear laboratory dc drive (AMIRA DR300). The hardware description of this system is described in the following section.

KHANESAR et al.: DIRECT MODEL REFERENCE TAKAGI–SUGENO FUZZY CONTROL OF SISO NONLINEAR SYSTEMS

Fig. 8.

Regulation control of Chua’s circuit.

Fig. 9. Square wave tracking performance of the controller when applied to Chua’s circuit.

A. Hardware Configuration of DR300 Servo DR300-AMIRA is a servomechanism consisting of one motor and a current controlled dc generator that are connected by a mechanical clutch (see Fig. 11). The dc motor is loaded using the dc generator mounted on its shaft. The nominal values of the parameters of the motors are given in Table II. The drive has built-in analog current controllers for both dc machines. The speed controller of the former machine is digitally implemented using an input–output card MF614 real-time control board. It runs the discrete controller with a sampling period that is equal to 10 ms. The two armature current references are fed to the dc drive using 16-bit D/A converters that are integrated in the MF614 board. The direct model reference fuzzy controller is written in MATLAB using the embedded function and runs on the host PC, in which all the configuration constants and the reference model are defined. B. Simplified Mathematical Description of DR300 Fig. 12 shows the electrical circuit diagram of the motor. The simplified mathematical representation of the system is derived

921

Fig. 10. Sinusoidal tracking performance of the controller when applied to Chua’s circuit.

Fig. 11.

Servo DR300-AMIRA setup.

Fig. 12.

Electrical diagram of the dc motor and the generator load.

as [28] ω ¨=−

1 1 1 UA ω˙ − ω+ TA TM TA TM TA CΦ

−

RA RA ML − M˙ L . KM TM TA CΦ KM TM CΦ

(51)

The aforementioned representation is stated to be very inaccurate by the manufacturers of the experimental setup [28]. The description of the parameters of this equation and their nominal values can be found in Tables I and II, respectively. It is possible to write the state-space model of the system as x˙ 1 = x2 x˙ 2 = − −

1 1 1 UA x1 − x2 + TM TA TA TM TA CΦ RA RA ML − M˙ L KM TM TA CΦ KM TM CΦ

(52)

˙ respectively. In the experin which x1 and x2 represent ω and ω, imental investigations, a state-dependent load that corresponds to the load–torque characteristics of a centrifugal pump is used

922

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

TABLE I NOMENCLATURE

TABLE II NOMINAL VALUES OF THE MACHINES

Fig. 13. Membership functions considered for the control of the dc motor with a nonlinear load.

as follows: T = k.ω 2

(53)

where T represents the load torque in (N · m), ω is the speed of the pump in (rad/s), and k is a constant in kg · m2 /rad2 . The load can, therefore, in this implementation, be considered as a function of the state x1 of the system, i.e., as ML = x21 . The simplified dynamical equation of the system can then be derived as x˙ 1 = x2 x˙ 2 = − −

1 1 1 UA x1 − x2 + TM TA TA TM TA CΦ RA 2RA x2 − x1 x2 . KM TM TA CΦ 1 KM TM CΦ

Fig. 14.

Tracking performance of the system.

Fig. 15.

Tracking performance of the system.

(54)

The desired eigenvalues for the system are selected as {−10, −100}. The reference model that correspond to these eigenvalues is given as   0 1 . Am = −1000 −110 Fig. 13 shows the membership functions used to control the dc motor. Fig. 14 shows the tracking performance of the dc motor and the reference signal. As can be seen from the figure, there is a very small error in the response of the system, caused by the uncertainties in the real time system and the modeling errors. As discussed earlier, in the presence of modeling errors, the Lyapunov function converges exponentially until V ≤ ρc , where ρ and c are defined in (35) and (36). In addition to show the effect of a sudden parameter change, the load is suddenly changed from ML = x21 to ML = 0.9x21 . Fig. 15 shows the

KHANESAR et al.: DIRECT MODEL REFERENCE TAKAGI–SUGENO FUZZY CONTROL OF SISO NONLINEAR SYSTEMS

tracking results of the system with a load change that happens at the 15th second. VI. CONCLUSION Generally, adaptive controllers can be considered in two categories: direct and indirect. Indirect adaptive controllers require an identifier and, therefore, suffer from high-computational cost. High-computational cost means, in real-time applications, longer delays between the sampling and the actual output of the control signal. To alleviate these problems, a direct model reference controller has been proposed in the form of a T–S fuzzy system. The main contribution of this paper with respect to the previous studies is that it has covered a more general class of T–S systems without the need for solving LMIs. With the current approach, the designer has more choices for the reference model. The effect of modeling errors in T–S systems has also been considered and the adaptation law has been modified to ensure robustness against parameter bursting. The proposed approach is simulated on two nonlinear plants. The first nonlinear plant is a flexible joint robot. After testing the tracking and the regulation performance of the system, the effectiveness of the robustness term is tested by turning off this term. The Frobenius norm of the parameters of the controller shows that without the robust term some parameter bursting may occur as a result of the modeling error. The next simulation example considered is a chaotic electrical circuit. It is shown that the proposed method no longer necessitates a feedback linearization form for the plant. Finally, the proposed method is implemented on a nonlinear dc motor with a nonlinear state-dependent load that depends on the square of the velocity of the motor. This type of load corresponds to the torque–velocity characteristics of centrifugal fans, pumps, and blowers in industrial applications. The effect of a small sudden change in the load (which is another type of modeling error) is also tested. It is, thus, shown that the proposed method is capable of controlling the nonlinear plants in the presence of plant-model mismatches. REFERENCES [1] J. Wang, A. B. Rad, and P. T. Chan, “Indirect adaptive fuzzy sliding mode control. Part I: Fuzzy switching,” Fuzzy Sets Syst., vol. 122, no. 1, pp. 21–30, 2001. [2] C.-F. Hsu and C.-M. Lin, “Fuzzy-identification-based adaptive controller design via backstepping approach,” Fuzzy Sets Syst., vol. 151, no. 1, pp. 43–57, 2005. [3] C.-W. Park and M. Park, “Adaptive parameter estimator based on T–S fuzzy models and its applications to indirect adaptive fuzzy control design,” Inf. Sci., vol. 159, no. 1–2, pp. 125–139, 2004. [4] K. J. Astrom and B. Wittenmark, Adaptive Control. New York: Dover, 2008. [5] Y.-W. Cho, C.-W. Park, and M. Park, “An indirect model reference adaptive fuzzy control for SISO Takagi–Sugeno model,” Fuzzy Sets Syst., vol. 131, no. 2, pp. 197–215, 2002. [6] Y.-W. Cho, C.-W. Park, J.-H. Kim, and M. Park, “Indirect model reference adaptive fuzzy control of dynamic fuzzy state-space model,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 148, no. 4, pp. 273–282, 2001. [7] T. Koo, “Stable model reference adaptive fuzzy control of a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 9, no. 4, pp. 624–636, Aug. 2001. [8] C.-W. Park and Y.-W. Cho, “Adaptive tracking control of flexible joint manipulator based on fuzzy model reference approach,” Proc. Inst. Elect. Eng. Control Theory Appl., vol. 150, no. 2, pp. 198–204, Mar. 2003.

923

[9] K. L. Y. Cho, Y. Lee, and E. Kim, “A Lyapunov function based direct model reference adaptive fuzzy control,” Knowl.-Based Intell. Inf. Eng. Syst., vol. 3214, pp. 202–210, 2004. [10] Y.-W. Cho, E.-S. Kim, K.-C. Lee, and M. Park, “Tracking control of a robot manipulator using a direct model reference adaptive fuzzy control,” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst., 1999, vol. 1, pp. 100–105. [11] M. A. Khanesar and M. Teshnehlab, “Direct stable adaptive fuzzy neural model reference control of a class of nonlinear systems,” in Proc. 3rd Int. Conf. Innovative Comput. Inf. Control, Jun. 2008, vol. 1, pp. 512 –515. [12] D.-L. Tsay, H.-Y. Chung, and C.-J. Lee, “The adaptive control of nonlinear systems using the Sugeno-type of fuzzy logic,” IEEE Trans. Fuzzy Syst., vol. 7, no. 2, pp. 225 –229, Apr. 1999. [13] S. Blazic, I. Skrjanc, and D. Matko, “Globally stable direct fuzzy model reference adaptive control,” Fuzzy Sets Syst., vol. 139, no. 1, pp. 3–33, 2003. [14] N. Golea, A. Golea, and K. Benmahammed, “Fuzzy model reference adaptive control,” IEEE Trans. Fuzzy Syst., vol. 10, no. 4, pp. 436–444, Aug. 2002. [15] W.-S. Yu and C.-J. Sun, “Fuzzy-model-based adaptive control for a class of nonlinear systems,” IEEE Trans. Fuzzy Syst., vol. 9, no. 3, pp. 413–425, Jun. 2001. [16] S. Kamalasadan and A. Ghandakly, “Multiple fuzzy reference model adaptive controller design for pitch-rate tracking,” IEEE Trans. Instrum. Meas., vol. 56, no. 5, pp. 1797–1808, Oct. 2007. [17] T. Takagi and M. Sugeno, “Fuzzy identification of systems and its applications to modeling and control,” IEEE Trans. Syst., Man, Cybern., vol. SMC-15, pp. 116–132, 1985. [18] J. A. Farrell and M. M. Polycarpou, Adaptive Approximation Based Control: Unifying Neural, Fuzzy and Traditional Adaptive Approximation Approaches. Hoboken, NJ: Wiley, 2006. [19] X.-J. Zeng, J. Keane, and D. Wang, “Fuzzy systems approach to approximation and stabilization of conventional affine nonlinear systems,” in Proc. IEEE Int. Conf. Fuzzy Syst., 2006, pp. 277–284. [20] B.-S. Chen, C.-S. Tseng, and H.-J. Uang, “Robustness design of nonlinear dynamic systems via fuzzy linear control,” IEEE Trans. Fuzzy Syst., vol. 7, no. 5, pp. 571–585, Oct. 1999. [21] P. A. Ioannou and J. Sun, Robust Adaptive Control. Englewood Cliffs, NJ: Prentice-Hall, 1996. [22] T. E. Krikochoritis and S. G. Tzafestas, “Control of flexible joint robots using neural networks,” IMA J. Math. Control Inf., vol. 18, no. 1, pp. 269– 280, 2001. [23] H. K. Khalil, Nonlinear Systems, 3rd ed. Englewood Cliffs, NJ: PrenticeHall, 2001. [24] C.-C. Hwang, H.-Y. Chow, and Y.-K. Wang, “A new feedback control of a modified Chua’s circuit system,” Physica D, vol. 92, pp. 95–100, 1996. [25] T.-L. Liao and F. W. Chen, “Control of Chua’s circuit with a cubic nonlinearity via nonlinear linearization technique,” Circuits Syst. Signal Process., vol. 17, no. 6, pp. 719–731, 1998. [26] M. Jang, C. Chen, and C. O. Chen, “Sliding mode control of chaos in the cubic Chua’s circuit system,” Int. J. Bifurcation Chaos, vol. 12, no. 6, pp. 1437–1449, 2002. [27] M. T. Yassen, “Adaptive control and synchronization of a modified Chua’s circuit system,” Appl. Math. Comput., vol. 135, pp. 113–128, 2003. [28] J. Roubal, P. Augusta, and V. Havlenca, “A brief introduction to control design demonstrated on laboratory model servo dr300 - amira,” Acta Electrotechnica et Informatica, vol. 5, no. 4, pp. 1–6, 2005.

Mojtaba Ahmadieh Khanesar (S’07) was born in Esfahan, Iran, on December 4, 1982. He received the B.Sc. and M.Sc. degrees in control engineering from the K. N. Toosi University of Technology, Tehran, Iran, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree in control engineering. His research interests include soft computing, intelligent control, adaptive control systems, fuzzy theory, and particle swarm optimization.

924

Okyay Kaynak (M’80–SM’90–F’03) received the B.Sc. (first-class Hons.) and Ph.D. degrees in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 1969 and 1972, respectively. From 1972 to 1979, he held various positions in the industry. In 1979, he joined the Department of Electrical and Electronics Engineering, Bogazici University, Istanbul, Turkey, where he is currently a Full Professor, holding the UNESCO Chair on Mechatronics. He has held long-term (near or more than a year) Visiting Professor/Scholar positions with various institutions in Japan, Germany, the United States, and Singapore. He is the author of three and the editor of five books. Additionally, he is the author or a coauthor of more than 300 papers that have appeared in various journals and conference proceedings. His current research interests include intelligent control and mechatronics. Dr. Kaynak is active in international organizations, has served on many committees of the IEEE, and was the President of the IEEE Industrial Electronics Society during 2002–2003.

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 19, NO. 5, OCTOBER 2011

Mohammad Teshnehlab received the B.Sc. degree in electrical engineering from Stony Brook University, Stony Brook, NY, in 1980, the Master’s degree in electrical engineering from Oita University, Oita, Japan, in 1991, and the Ph.D degree in computational intelligence from Saga University, Saga, Japan, in 1994. He is currently an Associate Professor of control systems with the Department of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran, Iran. He is the author or a coauthor of more than 130 papers that have appeared in various journals and conference proceedings. His main research interests include neural networks, fuzzy systems, evolutionary algorithms, swarm intelligence, fuzzy-neural networks, pattern recognition, metaheuristic algorithms and intelligent identification, and prediction and control.