Comments on" Output tracking and regulation of nonlinear system ...
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003
Fig. 13 shows the closed-loop response of a sinusoid function as the period is 4 s and the magnitude is 2. From these simulations, it is noted that the behavior of the linearizing plant is much like a linear system.
521
Comments on “Output Tracking and Regulation of Nonlinear System Based on Takagi–Sugeno Fuzzy Model” Ho Jae Lee, Jin Bae Park, and Young Hoon Joo
V. CONCLUSIONS The comparisons between the conventional backpropagating and proposed RLLS approaches to linearizing a nonlinear system have been presented. When the state variables of the plant are measured, the proposed controller can compute the linearizing control signal to efficiently linearize a nonlinear system. By a proper training process, the linearized controller can cancel the nonlinear terms of nonlinear system and yields any desired dynamics. The proposed training algorithm does not depend on plant modeling or its derivatives and is easy to implement. In other words, the proposed approach is a model-free method where there is no need to identify the dynamics of the plant in advance. Furthermore, it can solve one of the typical problems of linearization where the errors between the desired and regulated outcomes can hardly provide analytically instructive information, such as the gradients, for an adaptive system to change its behavior to improve performance accordingly, In the RLLS, the system learns the evaluative information, which is contained in reinforcement signals, to generate the predictive evaluation to direct whether improvement is possible or how the system should change. Since it has been utilizing the prediction-action control mechanism, the RLLS does not have to wait until the actual outcome is known and updates their parameters within the period without any evaluative feedback from the environments. After linearizing a nonlinear plant, the affine network-plant system can be controlled either by means of a PID controller, fuzzy controller, or by other modern control approaches. In other words, the linearized network-plant system can be easily controlled by means of rich control theories of linear systems. REFERENCES [1] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, pp. 4–27, Jan. 1990. [2] K. S. Hwang and H. J. Chao, “Adaptive reinforcement learning system for linearization control,” IEEE Trans. Ind. Electron., vol. 47, pp. 1185–1188, Oct. 2000. [3] M. H. R. Fazlur Rahman, R. Devanathan, and Z. Kuanyi, “Neural network approach for linearizing control of nonlinear process plants,” IEEE Trans. Ind. Electron., vol. 47, pp. 470–477, Apr. 2000. [4] C. W. Anderson, “Learning to control an inverted pendulum using neural networks,” IEEE Contr. Syst. Mag., pp. 31–37, 1989. [5] A. Y. Zomaya, “Reinforcement learning for the adaptive control of nonlinear systems,” IEEE Trans. Syst. Man, Cybern., vol. 24, pp. 357–363, Feb. 1994. [6] H. Shouling, K. Relf, and R. Unbehauen, “A neural approach for control of nonlinear systems with feedback linearization,” IEEE Trans. Neural Networks, vol. 9, pp. 1409–1421, Nov. 1998. [7] H. D. Patino and D. Liu, “Neural network-based model reference adaptive control system,” IEEE Trans. Syst. Man, Cybern. B, vol. 30, pp. 198–204, Feb. 2000. [8] A. Delgado and C. Kambhampati, “Relative degree of recurrent neural networks,” in Proc. Second Int. Conf. Intelligent Syst. Eng., 1994, pp. 113–117. [9] S. Yamada, M. Nakashima, and S. Shiono, “Reinforcement learning to train a cooperative network with both discrete and continuous output neurons,” IEEE Trans. Neural Networks, vol. 9, pp. 1502–1508, Nov. 1998. [10] R. S. Sutton, “Learning to predict by the methods of temporal differences,” Machine Learn., vol. 3, pp. 9–44, 1988. [11] V. Gullapalli, “A stochastic reinforcement learning algorithm for learning real-valued functions,” Neural Networks, vol. 3, pp. 671–692, 1990.
Abstract—In the above paper [1], a Takagi–Sugeno (T-S) fuzzy-model-based output regulation methodology is proposed. However, since the problem formulation is incorrect, the derived design techniques may not accomplish the control objectives. We illustrate the reasons and show counterexamples. Index Terms—Coordinate transformation, fuzzy control, output regulation.
I. INTRODUCTION The output stabilization of the Takagi–Sugeno (T–S) fuzzy model with disturbance is one of the most challenging, yet difficult problems in fuzzy control field. Recently, in [1] Ma et al. presented the output regulator design technique based on T-S fuzzy model via state feedback and error feedback in the presence of the known disturbance. In [1], the output regulation problems with the known disturbance were transformed into the regulation problem without disturbance using the simple coordinate transformation, which is widely used in tracking controller design for linear systems [2]–[4]. However, these reformulation is not correct, because the authors did not concerned the important fact that the T-S fuzzy model normally has highly nonlinear interaction among the plant rules and controller rules. Consequently, even if all conditions in [1, Th. 3.1 and 4.1] are satisfied, the control objectives, i.e., output regulation of the T-S fuzzy model, may not be achieved. In the next section, we describe the details. II. PROBLEMS OF THE PROPOSED CONTROL ALGORITHMS In this section, two cases are investigated: 1) output regulation via state feedback case and 2) output regulation via error feedback case. We now begin with the state feedback case. Problem 1: In [1, Th. 3.1], the authors proposed a T-S fuzzy-modelbased state feedback controller of the form
u(t) = 0
r i=1
i K i x(t) 0
r i=1
iLi w(t):
(1)
5
The gain matrices Li and the coordinate transformation matrices i were obtained from linear matrix equations (9) and (10) of [1]. With more precise observation of these equations, they represent the case that the ith local plant model is exclusively coupled with the ith local control law and ith disturbance model, not with the globally deffuzified ones. Here, it should be pointed out that they must not be the local but global ones which are inferred by the relevant fuzzy rules.
Manuscript received July 11, 2000. This paper was recommended by Associate Editor W. A. Gruver. H. J. Lee and J. B. Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]). Y. H. Joo is with the School of Electronic and Information Engineering, Kunsan National University, Chonbuk 573-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2003.811124
1083-4419/03$17.00 © 2003 IEEE
522
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003
Considering (1) and the coordinates transformation x ~i (t) = x(t) 0
5 w(t) for each plant rule i, we obtain i
x~_ i (t) = x_ (t) 0 5i w_ (t) = A i x(t)
0B
+ P i w (t)
r
j K j x(t) +
i j =1
0 5 S w(t) i
r j =1
j Lj w(t)
i
0 B K )x(t) + (P 0 B L 0 5 S )w(t) (K 0K )x(t)+ (L 0L )w(t) 0B
= (Ai
i
i
i
i
i
r
i
i
r
i
j
j
i
j
j =1
j
i
j =1
0 B K )x(t) 0 (A 0 B K )5 w(t) 0B (K 0K )x(t)+ (L 0L )w(t)
= (Ai
i
i
i
i
i
r
i
j
j
i
j
j =1
0 B K )(x(t) 0 5 w(t)) (K 0K )x(t)+ 0B
= (Ai
i
i
i
j
j
0 B K )~x (t) 0 B r
+ j =1
i
r
i
j =1
i
j
i
j =1
i
r
= (Ai
i
r
i
r i j =1
j =1
j (Lj 0Li )w(t)
Fig. 1.
Output regulation error e(t) of Example 1.
Fig. 2.
Output regulation error e(t) of Example 2.
j (K j 0 K i )x(t)
j (Lj 0 Li )w(t) :
(2)
Since the integration of (2) cannot yield
x~i (t) = exp((Ai 0 B iK i )t)~xi (0)
(3)
it is not clear that x ~i (t) converges to zero. Moreover,
!1 e (t) = lim !1 C x(t) + Q w(t) = lim C (~ !1 x (t) + 5 w(t)) + Q w(t) = lim C x !1 ~ (t) + lim !1 (C 5 + Q )w(t):
lim
t
i
t t t
i
i
i i
i
i
i
t
i
i
i
i
(4)
In the last equality of (4), the zero convergence of the second term is guaranteed from [1, Eq. (10)]. However, although K i are adequately designed by [1, Th. 1], the zero convergence of C i x ~i (t) is not assured, neither is the global error e(t). Therefore, the proposed control algorithm may not guarantee the output regulation of the T-S fuzzy model with the known disturbance. Problem 2: In [1, Th. 4.1], the following fuzzy model-based error feedback controller is used
u(t) =
r i=1
x~_ i (t) ~_ i (t)
Ai B iH i GiC i F i
=
Bi
iH i (t):
Using the coordinates transformations x ~i (t) = ~i (t) = (t) 0 0i w(t), we obtain
_
_ i (t) and ~i (t) should be modified the differential equation vectors for x ~ as
(5)
x(t) 0 5i w(t) and
x~_ i (t) = x_ (t) 0 5i w_ (t) = Ai x(t) + B i u(t) + P i w (t) 0 5i S i w (t) _ ~ i (t) = _(t) 0 0i w_ (t) = F i (t) + G i e(t) 0 0i S i w (t):
(6)
+
Gi
r j =1
r j =1
x~i (t) ~i (t) j (H j 0 H i ) (t)
j (C j 0 C i )x(t) +
r j =1
j (Qj 0 Qi )w(t)
: (8)
Therefore, the convergence to zero of the solution to (8) can not be claimed by [1, Th. 4.1] due to the last term in (8). This fact indicates that finding F i , Gi , H i and P by [1, Th. 4.1] may not guarantee the asymptotic stability of the output of the T-S fuzzy model with the known disturbance.
(7)
It also should be pointed out that the global control input u(t) and the global error e(t) should be plugged into (6) and (7), respectively, and
III. EXAMPLES In this section, two counterexamples are given to disprove [1, Th. 3.1 and 4.1]
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003
079 8822
523
060 4711 051 7722 030 1413 448 6743 0121 6682 0367 9849 163 5436 095 4072 170 0422 0112 2539 52 3858 70 6430 = 060 4711 448 6743 0112 2539 528 1573 0367 0261 0527 3036 051 7722 0121 6682 52 3858 0367 0261 651 8315 639 0372 030 1413 0367 9849 70 6430 0527 3036 639 0372 790 6842 8 5858 1 01 3572 0 22 3354 1 04 4671 0 0 147 9104 02 28 6621 0 04 0 86 9980 02 17 1996 0 4 1 = 2 = 137 9289 0 027 5858 1 206 6770 0 041 3354 1 97 9011 0 020 5802 0 184 9216 0 037 9843 0 = [ 0 8 5858 146 9104 0 137 9289 0 97 9011 ] = [ 0 22 3354 85 998 0 206 677 0184 9216 ] 1 2 1 = [ 92 3 011 6 04 176 0 232 ] 2 = [ 113 3048 023 2 0 4 64 ] 175:1795 :
:
:
:
:
:
:
F
:
:
:
:
:
:
:
51 = 00 36
:
0 0:20
:
:
:
:
:
:
:
:
:
:
:
:
;
;
H
0:2
0
0
0:2
B
0:1142 0:2701
00 3714 00 0115 1 = [ 07 4924 2 = [ 00 8718
P
=
:
:
K K
: :
00 3714 00 0115 00 2593 00 0305 :
:
:
:
00 2593 1 8341 0 0514 00 0305 0 0514 0 0279 011 3812 44 7500 11 ] 01 3243 4 9722 1 2222 ] :
:
:
:
:
:
:
:
:
;
:
:
L1
= [ 0:2508
1:2487
9:8114 ] ;
L2
= [ 0:0239
0:1453
1:1416 ]
00 1
1
0
0
0
1
0
0:1427
0
0
0:1427
1
0
0
0
1
0
0:1494
0
0
0
0:1494
:
51 =
0
00 1 :
52 =
;
:
However, as shown in Fig. 1, the global error e(t) does not converge to zero, which is expected from Problem 1. It is very difficult, if not impossible, to find i and Li such that (2) is asymptotically stable for all i except the special case. Consequently, the asymptotic stability of the output of the T-S fuzzy model is not guaranteed. Example 2: This counterexample illustrates Theorem 4.1 is erroneous. The minute change of the example in Section V-C is taken, that is, we redefine B 2 = [0 1=(2(M l2 + I ))]T and P 1 = 0:036 00 . The associated matrices which satisfy all conditions in [1, Th. 4.1] are found as shown in the equations at the top of the page. The simulation result in Fig. 2 shows that the global error does not converge to zero as time goes, which implies the asymptotic stability of the output of the T-S fuzzy model is not guaranteed, as is discussed in Problem 2.
5
IV. CONCLUSION It is difficult to design an output regulation controller for the T-S fuzzy model with the known disturbance because plant rules and controller rules of the T-S fuzzy model are highly interacted. Reference [1]
:
G
:
:
:
T
;
Example 1: The example in [1, Sec. V-B] is slightly modified. That is, only the input matrix 2 of the given example is redefined as [0 0 0 9]T . From [1, Th. 3.1], the associated matrices are found as follows: 0:4084
:
: :
52 =
0:1142
:
:
F
:
:
;
:
:
:
:
0:2
:
:
:
:
:
:
:
:
:
H
:
:
:
:
T
G
:
:
:
:
095 4072
560:6618
:
P
163:5436
:
079 8822
:
:
:
:
:
:
neglected this basic and essential fact in fuzzy-model-based control. It must be noted that the appropriate coordinate transformation matrices may not be obtained in the general T-S fuzzy model, accordingly the output stabilization conditions based on the coordinate transformation technique proposed in [1] do not hold. REFERENCES [1] X. J. Ma and Z. Q. Sun, “Output tracking and regulation of nonlinear system based on Takagi–Sugeno fuzzy model,” IEEE Trans. Syst., Man, Cybern. B, vol. 30, pp. 47–59, Feb. 2000. [2] T. H. Hopp and W. E. Schmitendorf, “Design of a linear controller for robust tracking and model following,” Trans. ASME, vol. 112, pp. 552–558, Dec. 1990. [3] W. E. Schmitendorf, Y. K. Kao, and H. Y. Hwang, “Robust tracking controller for a seeker scan loops control systems,” IEEE Trans. Control Syst. Technol., vol. 7, pp. 282–288, Mar. 1999. [4] S. Oucheriah, “Robust tracking and model following of uncertain dynamics delay systems by memoryless linear controllers,” IEEE Trans. Automat. Contr., vol. 44, pp. 1473–1477, July 1999.
Fig. 13 shows the closed-loop response of a sinusoid function as the period is 4 s and the magnitude is 2. From these simulations, it is noted that the behavior of the linearizing plant is much like a linear system.
521
Comments on “Output Tracking and Regulation of Nonlinear System Based on Takagi–Sugeno Fuzzy Model” Ho Jae Lee, Jin Bae Park, and Young Hoon Joo
V. CONCLUSIONS The comparisons between the conventional backpropagating and proposed RLLS approaches to linearizing a nonlinear system have been presented. When the state variables of the plant are measured, the proposed controller can compute the linearizing control signal to efficiently linearize a nonlinear system. By a proper training process, the linearized controller can cancel the nonlinear terms of nonlinear system and yields any desired dynamics. The proposed training algorithm does not depend on plant modeling or its derivatives and is easy to implement. In other words, the proposed approach is a model-free method where there is no need to identify the dynamics of the plant in advance. Furthermore, it can solve one of the typical problems of linearization where the errors between the desired and regulated outcomes can hardly provide analytically instructive information, such as the gradients, for an adaptive system to change its behavior to improve performance accordingly, In the RLLS, the system learns the evaluative information, which is contained in reinforcement signals, to generate the predictive evaluation to direct whether improvement is possible or how the system should change. Since it has been utilizing the prediction-action control mechanism, the RLLS does not have to wait until the actual outcome is known and updates their parameters within the period without any evaluative feedback from the environments. After linearizing a nonlinear plant, the affine network-plant system can be controlled either by means of a PID controller, fuzzy controller, or by other modern control approaches. In other words, the linearized network-plant system can be easily controlled by means of rich control theories of linear systems. REFERENCES [1] K. S. Narendra and K. Parthasarathy, “Identification and control of dynamical systems using neural networks,” IEEE Trans. Neural Networks, vol. 1, pp. 4–27, Jan. 1990. [2] K. S. Hwang and H. J. Chao, “Adaptive reinforcement learning system for linearization control,” IEEE Trans. Ind. Electron., vol. 47, pp. 1185–1188, Oct. 2000. [3] M. H. R. Fazlur Rahman, R. Devanathan, and Z. Kuanyi, “Neural network approach for linearizing control of nonlinear process plants,” IEEE Trans. Ind. Electron., vol. 47, pp. 470–477, Apr. 2000. [4] C. W. Anderson, “Learning to control an inverted pendulum using neural networks,” IEEE Contr. Syst. Mag., pp. 31–37, 1989. [5] A. Y. Zomaya, “Reinforcement learning for the adaptive control of nonlinear systems,” IEEE Trans. Syst. Man, Cybern., vol. 24, pp. 357–363, Feb. 1994. [6] H. Shouling, K. Relf, and R. Unbehauen, “A neural approach for control of nonlinear systems with feedback linearization,” IEEE Trans. Neural Networks, vol. 9, pp. 1409–1421, Nov. 1998. [7] H. D. Patino and D. Liu, “Neural network-based model reference adaptive control system,” IEEE Trans. Syst. Man, Cybern. B, vol. 30, pp. 198–204, Feb. 2000. [8] A. Delgado and C. Kambhampati, “Relative degree of recurrent neural networks,” in Proc. Second Int. Conf. Intelligent Syst. Eng., 1994, pp. 113–117. [9] S. Yamada, M. Nakashima, and S. Shiono, “Reinforcement learning to train a cooperative network with both discrete and continuous output neurons,” IEEE Trans. Neural Networks, vol. 9, pp. 1502–1508, Nov. 1998. [10] R. S. Sutton, “Learning to predict by the methods of temporal differences,” Machine Learn., vol. 3, pp. 9–44, 1988. [11] V. Gullapalli, “A stochastic reinforcement learning algorithm for learning real-valued functions,” Neural Networks, vol. 3, pp. 671–692, 1990.
Abstract—In the above paper [1], a Takagi–Sugeno (T-S) fuzzy-model-based output regulation methodology is proposed. However, since the problem formulation is incorrect, the derived design techniques may not accomplish the control objectives. We illustrate the reasons and show counterexamples. Index Terms—Coordinate transformation, fuzzy control, output regulation.
I. INTRODUCTION The output stabilization of the Takagi–Sugeno (T–S) fuzzy model with disturbance is one of the most challenging, yet difficult problems in fuzzy control field. Recently, in [1] Ma et al. presented the output regulator design technique based on T-S fuzzy model via state feedback and error feedback in the presence of the known disturbance. In [1], the output regulation problems with the known disturbance were transformed into the regulation problem without disturbance using the simple coordinate transformation, which is widely used in tracking controller design for linear systems [2]–[4]. However, these reformulation is not correct, because the authors did not concerned the important fact that the T-S fuzzy model normally has highly nonlinear interaction among the plant rules and controller rules. Consequently, even if all conditions in [1, Th. 3.1 and 4.1] are satisfied, the control objectives, i.e., output regulation of the T-S fuzzy model, may not be achieved. In the next section, we describe the details. II. PROBLEMS OF THE PROPOSED CONTROL ALGORITHMS In this section, two cases are investigated: 1) output regulation via state feedback case and 2) output regulation via error feedback case. We now begin with the state feedback case. Problem 1: In [1, Th. 3.1], the authors proposed a T-S fuzzy-modelbased state feedback controller of the form
u(t) = 0
r i=1
i K i x(t) 0
r i=1
iLi w(t):
(1)
5
The gain matrices Li and the coordinate transformation matrices i were obtained from linear matrix equations (9) and (10) of [1]. With more precise observation of these equations, they represent the case that the ith local plant model is exclusively coupled with the ith local control law and ith disturbance model, not with the globally deffuzified ones. Here, it should be pointed out that they must not be the local but global ones which are inferred by the relevant fuzzy rules.
Manuscript received July 11, 2000. This paper was recommended by Associate Editor W. A. Gruver. H. J. Lee and J. B. Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]). Y. H. Joo is with the School of Electronic and Information Engineering, Kunsan National University, Chonbuk 573-701, Korea (e-mail: [email protected]). Digital Object Identifier 10.1109/TSMCB.2003.811124
1083-4419/03$17.00 © 2003 IEEE
522
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003
Considering (1) and the coordinates transformation x ~i (t) = x(t) 0
5 w(t) for each plant rule i, we obtain i
x~_ i (t) = x_ (t) 0 5i w_ (t) = A i x(t)
0B
+ P i w (t)
r
j K j x(t) +
i j =1
0 5 S w(t) i
r j =1
j Lj w(t)
i
0 B K )x(t) + (P 0 B L 0 5 S )w(t) (K 0K )x(t)+ (L 0L )w(t) 0B
= (Ai
i
i
i
i
i
r
i
i
r
i
j
j
i
j
j =1
j
i
j =1
0 B K )x(t) 0 (A 0 B K )5 w(t) 0B (K 0K )x(t)+ (L 0L )w(t)
= (Ai
i
i
i
i
i
r
i
j
j
i
j
j =1
0 B K )(x(t) 0 5 w(t)) (K 0K )x(t)+ 0B
= (Ai
i
i
i
j
j
0 B K )~x (t) 0 B r
+ j =1
i
r
i
j =1
i
j
i
j =1
i
r
= (Ai
i
r
i
r i j =1
j =1
j (Lj 0Li )w(t)
Fig. 1.
Output regulation error e(t) of Example 1.
Fig. 2.
Output regulation error e(t) of Example 2.
j (K j 0 K i )x(t)
j (Lj 0 Li )w(t) :
(2)
Since the integration of (2) cannot yield
x~i (t) = exp((Ai 0 B iK i )t)~xi (0)
(3)
it is not clear that x ~i (t) converges to zero. Moreover,
!1 e (t) = lim !1 C x(t) + Q w(t) = lim C (~ !1 x (t) + 5 w(t)) + Q w(t) = lim C x !1 ~ (t) + lim !1 (C 5 + Q )w(t):
lim
t
i
t t t
i
i
i i
i
i
i
t
i
i
i
i
(4)
In the last equality of (4), the zero convergence of the second term is guaranteed from [1, Eq. (10)]. However, although K i are adequately designed by [1, Th. 1], the zero convergence of C i x ~i (t) is not assured, neither is the global error e(t). Therefore, the proposed control algorithm may not guarantee the output regulation of the T-S fuzzy model with the known disturbance. Problem 2: In [1, Th. 4.1], the following fuzzy model-based error feedback controller is used
u(t) =
r i=1
x~_ i (t) ~_ i (t)
Ai B iH i GiC i F i
=
Bi
iH i (t):
Using the coordinates transformations x ~i (t) = ~i (t) = (t) 0 0i w(t), we obtain
_
_ i (t) and ~i (t) should be modified the differential equation vectors for x ~ as
(5)
x(t) 0 5i w(t) and
x~_ i (t) = x_ (t) 0 5i w_ (t) = Ai x(t) + B i u(t) + P i w (t) 0 5i S i w (t) _ ~ i (t) = _(t) 0 0i w_ (t) = F i (t) + G i e(t) 0 0i S i w (t):
(6)
+
Gi
r j =1
r j =1
x~i (t) ~i (t) j (H j 0 H i ) (t)
j (C j 0 C i )x(t) +
r j =1
j (Qj 0 Qi )w(t)
: (8)
Therefore, the convergence to zero of the solution to (8) can not be claimed by [1, Th. 4.1] due to the last term in (8). This fact indicates that finding F i , Gi , H i and P by [1, Th. 4.1] may not guarantee the asymptotic stability of the output of the T-S fuzzy model with the known disturbance.
(7)
It also should be pointed out that the global control input u(t) and the global error e(t) should be plugged into (6) and (7), respectively, and
III. EXAMPLES In this section, two counterexamples are given to disprove [1, Th. 3.1 and 4.1]
IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 33, NO. 3, JUNE 2003
079 8822
523
060 4711 051 7722 030 1413 448 6743 0121 6682 0367 9849 163 5436 095 4072 170 0422 0112 2539 52 3858 70 6430 = 060 4711 448 6743 0112 2539 528 1573 0367 0261 0527 3036 051 7722 0121 6682 52 3858 0367 0261 651 8315 639 0372 030 1413 0367 9849 70 6430 0527 3036 639 0372 790 6842 8 5858 1 01 3572 0 22 3354 1 04 4671 0 0 147 9104 02 28 6621 0 04 0 86 9980 02 17 1996 0 4 1 = 2 = 137 9289 0 027 5858 1 206 6770 0 041 3354 1 97 9011 0 020 5802 0 184 9216 0 037 9843 0 = [ 0 8 5858 146 9104 0 137 9289 0 97 9011 ] = [ 0 22 3354 85 998 0 206 677 0184 9216 ] 1 2 1 = [ 92 3 011 6 04 176 0 232 ] 2 = [ 113 3048 023 2 0 4 64 ] 175:1795 :
:
:
:
:
:
:
F
:
:
:
:
:
:
:
51 = 00 36
:
0 0:20
:
:
:
:
:
:
:
:
:
:
:
:
;
;
H
0:2
0
0
0:2
B
0:1142 0:2701
00 3714 00 0115 1 = [ 07 4924 2 = [ 00 8718
P
=
:
:
K K
: :
00 3714 00 0115 00 2593 00 0305 :
:
:
:
00 2593 1 8341 0 0514 00 0305 0 0514 0 0279 011 3812 44 7500 11 ] 01 3243 4 9722 1 2222 ] :
:
:
:
:
:
:
:
:
;
:
:
L1
= [ 0:2508
1:2487
9:8114 ] ;
L2
= [ 0:0239
0:1453
1:1416 ]
00 1
1
0
0
0
1
0
0:1427
0
0
0:1427
1
0
0
0
1
0
0:1494
0
0
0
0:1494
:
51 =
0
00 1 :
52 =
;
:
However, as shown in Fig. 1, the global error e(t) does not converge to zero, which is expected from Problem 1. It is very difficult, if not impossible, to find i and Li such that (2) is asymptotically stable for all i except the special case. Consequently, the asymptotic stability of the output of the T-S fuzzy model is not guaranteed. Example 2: This counterexample illustrates Theorem 4.1 is erroneous. The minute change of the example in Section V-C is taken, that is, we redefine B 2 = [0 1=(2(M l2 + I ))]T and P 1 = 0:036 00 . The associated matrices which satisfy all conditions in [1, Th. 4.1] are found as shown in the equations at the top of the page. The simulation result in Fig. 2 shows that the global error does not converge to zero as time goes, which implies the asymptotic stability of the output of the T-S fuzzy model is not guaranteed, as is discussed in Problem 2.
5
IV. CONCLUSION It is difficult to design an output regulation controller for the T-S fuzzy model with the known disturbance because plant rules and controller rules of the T-S fuzzy model are highly interacted. Reference [1]
:
G
:
:
:
T
;
Example 1: The example in [1, Sec. V-B] is slightly modified. That is, only the input matrix 2 of the given example is redefined as [0 0 0 9]T . From [1, Th. 3.1], the associated matrices are found as follows: 0:4084
:
: :
52 =
0:1142
:
:
F
:
:
;
:
:
:
:
0:2
:
:
:
:
:
:
:
:
:
H
:
:
:
:
T
G
:
:
:
:
095 4072
560:6618
:
P
163:5436
:
079 8822
:
:
:
:
:
:
neglected this basic and essential fact in fuzzy-model-based control. It must be noted that the appropriate coordinate transformation matrices may not be obtained in the general T-S fuzzy model, accordingly the output stabilization conditions based on the coordinate transformation technique proposed in [1] do not hold. REFERENCES [1] X. J. Ma and Z. Q. Sun, “Output tracking and regulation of nonlinear system based on Takagi–Sugeno fuzzy model,” IEEE Trans. Syst., Man, Cybern. B, vol. 30, pp. 47–59, Feb. 2000. [2] T. H. Hopp and W. E. Schmitendorf, “Design of a linear controller for robust tracking and model following,” Trans. ASME, vol. 112, pp. 552–558, Dec. 1990. [3] W. E. Schmitendorf, Y. K. Kao, and H. Y. Hwang, “Robust tracking controller for a seeker scan loops control systems,” IEEE Trans. Control Syst. Technol., vol. 7, pp. 282–288, Mar. 1999. [4] S. Oucheriah, “Robust tracking and model following of uncertain dynamics delay systems by memoryless linear controllers,” IEEE Trans. Automat. Contr., vol. 44, pp. 1473–1477, July 1999.
