- Email: [email protected]
Discussion on: “Robust H∞ Control for Uncertain Fuzzy Neutral Delay Systems”
382
Discussion on: ‘‘Robust H1 Control for Fuzzy Neutral Systems’’
of Ahi and the system (2) is rewritten as Ex_ e ðtÞ ¼
r X
( hi ðsðtÞÞ A^i xe ðtÞ A^mi
Z
i¼1
delay-dependent conditions from such a system make the stability regions much wider. 0
1
xðt þ sÞds )
i wðtÞ , þ A^hi xðt 1 Þ þ Adi xðt 2 Þ þ D ð3Þ where " A^i ¼ A^mi
0
A i þ Mi " # 0 ¼ : Mi
I I
"
# ,
A^hi ¼
0
#
Ahi Mi
,
Mi generalizes the system representation. In fact, if Mi ¼ Ahi for all i, then the system (3) reduces to (2). Similar to those for the system (2), delay-dependent conditions for the system (3) can be obtained with Mi as free parameter matrices. As is easily seen, Mi can be chosen to satisfy delay-dependent conditions, which reduce conservativeness and make the stability region wider. The above technique can be applied to a class of T–S fuzzy neutral systems with uncertainties in the parameters. Consequently, we shall have similar results for robust stability and robust H1 control. Moreover, robust stabilization and robust H1 control design can be solved with such relaxed conditions. A final remark goes to more relaxed conditions [10]. When an arbitrary state equation is added to the system (2), it has more freedom. The resulting
References 1. Fridman E, Shaked U. A descriptor system approach to H1 control of linear time-delay systems. IEEE Trans Automat Control 2002; 47: 253–270 2. Mahmoud MS. Robust control and filtering for time-delay systems. Marcel Dekker, Inc., New York, 2000 3. Mahmoud MS, Al-Muthairi NF. Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties. IEEE Tran Automat Control 1994; 39: 2135–2139 4. Park PG. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans Automat Control 1999; 44: 876–877 5. Xie L, de Souza CE. Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach. IEEE Trans Automat Control 1997; 42: 1144–1148 6. Yoneyama J. Robust control analysis for uncertain fuzzy systems with time-delays. In: Proceedings of the Joint 1st International Conference on Soft Computing and Intelligent Systems and 3rd International Symposium on Advanced Intelligent Systems (SCIS & ISIS 2002), 22P5–2, Tsukuba, Japan, October 2002 7. Yoneyama J. Stability and stabilization of fuzzy timedelay systems. In: Proceedings of the 5th Asian Control Conference, Melbourne, Australia, July 2004 (to be presented) 8. Yoneyama J. H control for fuzzy time-delay systems via descriptor approach. In: IEEE International Symposium on Intelligent Control, Taipei, Taiwan, September 2004 (to be presented) 9. Yoneyama J. Design of H-control for fuzzy timedelay systems. Fuzzy Sets and Systems, 2004 (to appear) 10. Yoneyama J. More relaxed conditions for Takagi– Sugeno fuzzy time-delay systems (in preparation)
Discussion on: ‘‘Robust H1 Control for Uncertain Fuzzy Neutral Delay Systems’’ Eun Tae Jeung1, and Hong Bae Park2, 1
Department of Control and Instrumentation Engineering, Changwon National University, 9 Sarimdong, Changwon, Kyungnam 641-773, Republic of Korea; 2School of Electrical Engineering and Computer Science, Kyungpook National University, Daegu, 702-701, Republic of Korea
In this paper, the authors have dealt with robust H1 control problem for a class of E-mail: [email protected] E-mail: [email protected]
Takagi–Sugeno (T–S) fuzzy systems with constant time delays of neutral type and norm-bounded time-varying parameter uncertainties. This paper is well organized and clearly written to obtain the results.
383
Discussion on: ‘‘Robust H1 Control for Fuzzy Neutral Systems’’
The paper has some minor errors as follows: (1) In the 7th line in the introduction on p. 1, ‘nonlinear’ must be replaced by ‘linear’. (2) In the 13th line in the problem formulation on p. 3, the authors said ‘it is assumed that the premise variables do not depend on the input variables u(t) explicitly’. The assumption was not used anywhere in the paper, so the sentence should be deleted. (3) In Theorems 2 and 4, Yi was missed out. (4) In the 2nd line above Eq. (55), w 2 L2 ½0, 1Þ must be replaced by w 2 L2 ½0, T because the integration in (55) is from 0 to T. In the same manner, the space of L2 ½0, 1Þ should explain the space ofL2 ½0, T. The paper has dealt with the case of delayindependent problem. The authors said in Remark 2 that delay-independent stability and stabilization can be available to the case with no prior knowledge about the size of the delay. Although we do not know the size of the delay, it is bounded in many practical systems. Delay-dependent analysis has received considerable attention and has been one of the most interesting topics in the control. Recently, delay-dependent control problem for time-delayed T–S fuzzy systems was investigated by Jeung et al. [1]. A sufficient condition for the existence of a PDC-type controller was
represented in terms of LMIs. A delay-dependent condition can be obtained from the extension of [1] to fuzzy neutral delay systems. Delay-dependent stability criteria for neutral delay systems was proposed by Park and Kwon [2] and He et al. [3]. In order to get less conservative R t conditions, they used the operators, Dðxt Þ ¼ xðtÞþ t GxðsÞds Ad xðtÞ and Dðxt Þ ¼ xðtÞAd xðtÞ, respectively. Extending the results of [2] and [3] to a control problem for T–S fuzzy systems, a delay-dependent control for T–S fuzzy neutral systems with uncertainties can be considered.
References 1. Jeung ET, Oh DC, Park HB. Delay-dependent control for time-delayed T–S fuzzy systems using descriptor representation. Int J Control, Automat Syst 2004; 2(2): 182–188 2. Park JH, Kwon O. On new stability criterion for delaydifferential systems of neutral type. Appl Math Comput available online 1 March 2004 (in press) 3. He Y, Wu M, She J-H, Liu GP. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Control Lett 2004; 51: 57–65
Discussion on: ‘‘Robust H1 Control for Uncertain Fuzzy Neutral Delay Systems’’ Yong-Yan Cao Institute of Modern Control Engineering, Department of Control Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China
Nonlinear systems with time-delay constitute basic mathematical models of real phenomena, for instance in circuits theory, economics and mechanics. Not only are dynamic systems with time-delay common in chemical processes and long-transmission lines in pneumatic, hydraulic, or rolling mill systems, but also computer controlled systems requiring numerical computation which have time-delays in control loops. The presence of time-delays in control loops usually degrades system performance and complicates the
E-mail: [email protected]
analysis and design of feedback controllers. Stability analysis and synthesis of retarded systems is an important issue in the area of control theory and application. The paper by Xu et al. presents an interesting design method on the robust H1 control for a class of uncertain Takagi–Sugeno (T–S) fuzzy neutral systems with constant time-delay and norm-bounded time-varying parameter uncertainties. Sufficient condition on the stability of the uncertain fuzzy neutral systems is obtained in terms of linear matrix inequalities. It is known that the T–S model provides an effective way to represent complex systems
Discussion on: ‘‘Robust H1 Control for Fuzzy Neutral Systems’’
of Ahi and the system (2) is rewritten as Ex_ e ðtÞ ¼
r X
( hi ðsðtÞÞ A^i xe ðtÞ A^mi
Z
i¼1
delay-dependent conditions from such a system make the stability regions much wider. 0
1
xðt þ sÞds )
i wðtÞ , þ A^hi xðt 1 Þ þ Adi xðt 2 Þ þ D ð3Þ where " A^i ¼ A^mi
0
A i þ Mi " # 0 ¼ : Mi
I I
"
# ,
A^hi ¼
0
#
Ahi Mi
,
Mi generalizes the system representation. In fact, if Mi ¼ Ahi for all i, then the system (3) reduces to (2). Similar to those for the system (2), delay-dependent conditions for the system (3) can be obtained with Mi as free parameter matrices. As is easily seen, Mi can be chosen to satisfy delay-dependent conditions, which reduce conservativeness and make the stability region wider. The above technique can be applied to a class of T–S fuzzy neutral systems with uncertainties in the parameters. Consequently, we shall have similar results for robust stability and robust H1 control. Moreover, robust stabilization and robust H1 control design can be solved with such relaxed conditions. A final remark goes to more relaxed conditions [10]. When an arbitrary state equation is added to the system (2), it has more freedom. The resulting
References 1. Fridman E, Shaked U. A descriptor system approach to H1 control of linear time-delay systems. IEEE Trans Automat Control 2002; 47: 253–270 2. Mahmoud MS. Robust control and filtering for time-delay systems. Marcel Dekker, Inc., New York, 2000 3. Mahmoud MS, Al-Muthairi NF. Quadratic stabilization of continuous time systems with state-delay and norm-bounded time-varying uncertainties. IEEE Tran Automat Control 1994; 39: 2135–2139 4. Park PG. A delay-dependent stability criterion for systems with uncertain time-invariant delays. IEEE Trans Automat Control 1999; 44: 876–877 5. Xie L, de Souza CE. Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach. IEEE Trans Automat Control 1997; 42: 1144–1148 6. Yoneyama J. Robust control analysis for uncertain fuzzy systems with time-delays. In: Proceedings of the Joint 1st International Conference on Soft Computing and Intelligent Systems and 3rd International Symposium on Advanced Intelligent Systems (SCIS & ISIS 2002), 22P5–2, Tsukuba, Japan, October 2002 7. Yoneyama J. Stability and stabilization of fuzzy timedelay systems. In: Proceedings of the 5th Asian Control Conference, Melbourne, Australia, July 2004 (to be presented) 8. Yoneyama J. H control for fuzzy time-delay systems via descriptor approach. In: IEEE International Symposium on Intelligent Control, Taipei, Taiwan, September 2004 (to be presented) 9. Yoneyama J. Design of H-control for fuzzy timedelay systems. Fuzzy Sets and Systems, 2004 (to appear) 10. Yoneyama J. More relaxed conditions for Takagi– Sugeno fuzzy time-delay systems (in preparation)
Discussion on: ‘‘Robust H1 Control for Uncertain Fuzzy Neutral Delay Systems’’ Eun Tae Jeung1, and Hong Bae Park2, 1
Department of Control and Instrumentation Engineering, Changwon National University, 9 Sarimdong, Changwon, Kyungnam 641-773, Republic of Korea; 2School of Electrical Engineering and Computer Science, Kyungpook National University, Daegu, 702-701, Republic of Korea
In this paper, the authors have dealt with robust H1 control problem for a class of E-mail: [email protected] E-mail: [email protected]
Takagi–Sugeno (T–S) fuzzy systems with constant time delays of neutral type and norm-bounded time-varying parameter uncertainties. This paper is well organized and clearly written to obtain the results.
383
Discussion on: ‘‘Robust H1 Control for Fuzzy Neutral Systems’’
The paper has some minor errors as follows: (1) In the 7th line in the introduction on p. 1, ‘nonlinear’ must be replaced by ‘linear’. (2) In the 13th line in the problem formulation on p. 3, the authors said ‘it is assumed that the premise variables do not depend on the input variables u(t) explicitly’. The assumption was not used anywhere in the paper, so the sentence should be deleted. (3) In Theorems 2 and 4, Yi was missed out. (4) In the 2nd line above Eq. (55), w 2 L2 ½0, 1Þ must be replaced by w 2 L2 ½0, T because the integration in (55) is from 0 to T. In the same manner, the space of L2 ½0, 1Þ should explain the space ofL2 ½0, T. The paper has dealt with the case of delayindependent problem. The authors said in Remark 2 that delay-independent stability and stabilization can be available to the case with no prior knowledge about the size of the delay. Although we do not know the size of the delay, it is bounded in many practical systems. Delay-dependent analysis has received considerable attention and has been one of the most interesting topics in the control. Recently, delay-dependent control problem for time-delayed T–S fuzzy systems was investigated by Jeung et al. [1]. A sufficient condition for the existence of a PDC-type controller was
represented in terms of LMIs. A delay-dependent condition can be obtained from the extension of [1] to fuzzy neutral delay systems. Delay-dependent stability criteria for neutral delay systems was proposed by Park and Kwon [2] and He et al. [3]. In order to get less conservative R t conditions, they used the operators, Dðxt Þ ¼ xðtÞþ t GxðsÞds Ad xðtÞ and Dðxt Þ ¼ xðtÞAd xðtÞ, respectively. Extending the results of [2] and [3] to a control problem for T–S fuzzy systems, a delay-dependent control for T–S fuzzy neutral systems with uncertainties can be considered.
References 1. Jeung ET, Oh DC, Park HB. Delay-dependent control for time-delayed T–S fuzzy systems using descriptor representation. Int J Control, Automat Syst 2004; 2(2): 182–188 2. Park JH, Kwon O. On new stability criterion for delaydifferential systems of neutral type. Appl Math Comput available online 1 March 2004 (in press) 3. He Y, Wu M, She J-H, Liu GP. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Control Lett 2004; 51: 57–65
Discussion on: ‘‘Robust H1 Control for Uncertain Fuzzy Neutral Delay Systems’’ Yong-Yan Cao Institute of Modern Control Engineering, Department of Control Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China
Nonlinear systems with time-delay constitute basic mathematical models of real phenomena, for instance in circuits theory, economics and mechanics. Not only are dynamic systems with time-delay common in chemical processes and long-transmission lines in pneumatic, hydraulic, or rolling mill systems, but also computer controlled systems requiring numerical computation which have time-delays in control loops. The presence of time-delays in control loops usually degrades system performance and complicates the
E-mail: [email protected]
analysis and design of feedback controllers. Stability analysis and synthesis of retarded systems is an important issue in the area of control theory and application. The paper by Xu et al. presents an interesting design method on the robust H1 control for a class of uncertain Takagi–Sugeno (T–S) fuzzy neutral systems with constant time-delay and norm-bounded time-varying parameter uncertainties. Sufficient condition on the stability of the uncertain fuzzy neutral systems is obtained in terms of linear matrix inequalities. It is known that the T–S model provides an effective way to represent complex systems